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Theorem constr3trllem2 26179
 Description: Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
Assertion
Ref Expression
constr3trllem2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)

Proof of Theorem constr3trllem2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 constr3cycl.f . . . . . 6 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
2 constr3cycl.p . . . . . 6 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
31, 2constr3trllem1 26178 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)
4 wrdf 13165 . . . . 5 (𝐹 ∈ Word dom 𝐸𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
53, 4syl 17 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
61, 2constr3lem2 26174 . . . . 5 (#‘𝐹) = 3
7 usgraf1o 25887 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
8 3cycl3dv 26170 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐴𝐵𝐵𝐶𝐶𝐴))
9 usgraedgrnv 25906 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
109ex 449 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝐴𝑉𝐵𝑉)))
11 usgraedgrnv 25906 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐶𝑉𝐴𝑉))
1211ex 449 . . . . . . . . . . . . . . . . . . 19 (𝑉 USGrph 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 → (𝐶𝑉𝐴𝑉)))
1310, 12anim12d 584 . . . . . . . . . . . . . . . . . 18 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
1413com12 32 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
15143adant2 1073 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉))))
1615impcom 445 . . . . . . . . . . . . . . 15 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)))
171, 2constr3lem5 26176 . . . . . . . . . . . . . . . . 17 ((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴}))
1817jctl 562 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
1918exp31 628 . . . . . . . . . . . . . . 15 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
2016, 19mpancom 700 . . . . . . . . . . . . . 14 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
218, 20mpd 15 . . . . . . . . . . . . 13 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))))
2221ex 449 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))))
237, 22mpid 43 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))))
2423imp 444 . . . . . . . . . 10 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
2524adantl 481 . . . . . . . . 9 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)))
26 c0ex 9913 . . . . . . . . . . 11 0 ∈ V
27 1ex 9914 . . . . . . . . . . 11 1 ∈ V
28 2z 11286 . . . . . . . . . . 11 2 ∈ ℤ
2926, 27, 283pm3.2i 1232 . . . . . . . . . 10 (0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ)
30 eqidd 2611 . . . . . . . . . . . . . . 15 ((𝐹‘0) = (𝐹‘0) → 0 = 0)
3130a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘0) → 0 = 0))
32 f1of1 6049 . . . . . . . . . . . . . . . . . . 19 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
3332adantl 481 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → 𝐸:dom 𝐸1-1→ran 𝐸)
34 3simpa 1051 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
3534adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
3635ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
37 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
3833, 36, 37syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
39 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐴𝑉𝐵𝑉))
40 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴𝑉𝐵𝑉) → 𝐵𝑉)
41 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐶𝑉𝐴𝑉) → 𝐶𝑉)
4240, 41anim12i 588 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐵𝑉𝐶𝑉))
43 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
4439, 42, 43syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
45 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
46 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 0 = 1))
4745, 46sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 0 = 1))
48473ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 0 = 1))
4948com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5049adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐵𝐵 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
51 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐶𝐴 ↔ ¬ 𝐶 = 𝐴)
52 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐶 = 𝐴 → (𝐶 = 𝐴 → 0 = 1))
5351, 52sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐶𝐴 → (𝐶 = 𝐴 → 0 = 1))
54533ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 0 = 1))
5554com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5655eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5756adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐶𝐵 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5850, 57jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1))
5944, 58syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 1)))
6059com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1)))
6160adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1)))
6261imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1))
6362adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 0 = 1))
6438, 63syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1))
6564adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1))
66 eqeq12 2623 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘0) = (𝐹‘1) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶})))
67663adant3 1074 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘1) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶})))
6867imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘0) = (𝐹‘1) → 0 = 1) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1)))
6968adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘1) → 0 = 1) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 0 = 1)))
7065, 69mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘1) → 0 = 1))
71 3simpb 1052 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
7271adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
7372ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
74 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → {𝐴, 𝐵} = {𝐶, 𝐴}))
7533, 73, 74syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → {𝐴, 𝐵} = {𝐶, 𝐴}))
76 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐶, 𝐴} ↔ ((𝐴 = 𝐶𝐵 = 𝐴) ∨ (𝐴 = 𝐴𝐵 = 𝐶))))
77 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 0 = 2))
7845, 77sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 0 = 2))
79783ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 0 = 2))
8079com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8180eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8281adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐶𝐵 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
83 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
84 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐵 = 𝐶 → (𝐵 = 𝐶 → 0 = 2))
8583, 84sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵𝐶 → (𝐵 = 𝐶 → 0 = 2))
86853ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 0 = 2))
8786com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8887adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 = 𝐴𝐵 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
8982, 88jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴 = 𝐶𝐵 = 𝐴) ∨ (𝐴 = 𝐴𝐵 = 𝐶)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2))
9076, 89syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐴, 𝐵} = {𝐶, 𝐴} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 0 = 2)))
9190com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2)))
9291adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2)))
9392imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2))
9493adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐴, 𝐵} = {𝐶, 𝐴} → 0 = 2))
9575, 94syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2))
9695adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2))
97 eqeq12 2623 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘2) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴})))
98973adant2 1073 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘0) = (𝐹‘2) ↔ (𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴})))
9998imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘0) = (𝐹‘2) → 0 = 2) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2)))
10099adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘2) → 0 = 2) ↔ ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐶, 𝐴}) → 0 = 2)))
10196, 100mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘0) = (𝐹‘2) → 0 = 2))
10231, 70, 1013jca 1235 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
103102adantl 481 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
104 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
105104eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘0)))
106 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (0 = 𝑦 ↔ 0 = 0))
107105, 106imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘0) → 0 = 0)))
108 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑦 = 1 → (𝐹𝑦) = (𝐹‘1))
109108eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘1)))
110 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (0 = 𝑦 ↔ 0 = 1))
111109, 110imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘1) → 0 = 1)))
112 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑦 = 2 → (𝐹𝑦) = (𝐹‘2))
113112eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘0) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹‘2)))
114 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (0 = 𝑦 ↔ 0 = 2))
115113, 114imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ ((𝐹‘0) = (𝐹‘2) → 0 = 2)))
116107, 111, 115raltpg 4183 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2))))
117116adantr 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ↔ (((𝐹‘0) = (𝐹‘0) → 0 = 0) ∧ ((𝐹‘0) = (𝐹‘1) → 0 = 1) ∧ ((𝐹‘0) = (𝐹‘2) → 0 = 2))))
118103, 117mpbird 246 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦))
119 pm3.22 464 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
1201193adant3 1074 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
121120adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
122121ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
123 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → {𝐵, 𝐶} = {𝐴, 𝐵}))
12433, 122, 123syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → {𝐵, 𝐶} = {𝐴, 𝐵}))
125 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵𝑉𝐶𝑉) ∧ (𝐴𝑉𝐵𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} ↔ ((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴))))
12642, 39, 125syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} ↔ ((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴))))
127 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 1 = 0))
12845, 127sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 1 = 0))
1291283ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 1 = 0))
130129com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
131130eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
132131adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐴𝐶 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
133 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐶 = 𝐴 → (𝐶 = 𝐴 → 1 = 0))
13451, 133sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐶𝐴 → (𝐶 = 𝐴 → 1 = 0))
1351343ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 1 = 0))
136135com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
137136adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐵𝐶 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
138132, 137jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 = 𝐴𝐶 = 𝐵) ∨ (𝐵 = 𝐵𝐶 = 𝐴)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0))
139126, 138syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐴, 𝐵} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 0)))
140139com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0)))
141140adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0)))
142141imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0))
143142adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} = {𝐴, 𝐵} → 1 = 0))
144124, 143syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0))
145144adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0))
146 eqeq12 2623 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘0) = (𝐸‘{𝐴, 𝐵})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
147146ancoms 468 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
1481473adant3 1074 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘0) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵})))
149148imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0)))
150149adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐴, 𝐵}) → 1 = 0)))
151145, 150mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘0) → 1 = 0))
152 eqidd 2611 . . . . . . . . . . . . . . 15 ((𝐹‘1) = (𝐹‘1) → 1 = 1)
153152a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘1) → 1 = 1))
154 3simpc 1053 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
155154adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
156155ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
157 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → {𝐵, 𝐶} = {𝐶, 𝐴}))
15833, 156, 157syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → {𝐵, 𝐶} = {𝐶, 𝐴}))
159 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵𝑉𝐶𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} ↔ ((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶))))
16042, 159sylancom 698 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} ↔ ((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶))))
161 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐵 = 𝐶 → (𝐵 = 𝐶 → 1 = 2))
16283, 161sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵𝐶 → (𝐵 = 𝐶 → 1 = 2))
1631623ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 1 = 2))
164163com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
165164adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐶𝐶 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
166 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐴 = 𝐵 → (𝐴 = 𝐵 → 1 = 2))
16745, 166sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐴𝐵 → (𝐴 = 𝐵 → 1 = 2))
1681673ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 1 = 2))
169168com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
170169eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐵 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
171170adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐵 = 𝐴𝐶 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
172165, 171jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐵 = 𝐶𝐶 = 𝐴) ∨ (𝐵 = 𝐴𝐶 = 𝐶)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2))
173160, 172syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐵, 𝐶} = {𝐶, 𝐴} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 1 = 2)))
174173com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2)))
175174adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2)))
176175imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2))
177176adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐵, 𝐶} = {𝐶, 𝐴} → 1 = 2))
178158, 177syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2))
179178adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2))
180 eqeq12 2623 . . . . . . . . . . . . . . . . . 18 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘2) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴})))
1811803adant1 1072 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘1) = (𝐹‘2) ↔ (𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴})))
182181imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘1) = (𝐹‘2) → 1 = 2) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2)))
183182adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘2) → 1 = 2) ↔ ((𝐸‘{𝐵, 𝐶}) = (𝐸‘{𝐶, 𝐴}) → 1 = 2)))
184179, 183mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘1) = (𝐹‘2) → 1 = 2))
185151, 153, 1843jca 1235 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
186185adantl 481 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
187104eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘0)))
188 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (1 = 𝑦 ↔ 1 = 0))
189187, 188imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘0) → 1 = 0)))
190108eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘1)))
191 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (1 = 𝑦 ↔ 1 = 1))
192190, 191imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘1) → 1 = 1)))
193112eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘1) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹‘2)))
194 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (1 = 𝑦 ↔ 1 = 2))
195193, 194imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ ((𝐹‘1) = (𝐹‘2) → 1 = 2)))
196189, 192, 195raltpg 4183 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2))))
197196adantr 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ↔ (((𝐹‘1) = (𝐹‘0) → 1 = 0) ∧ ((𝐹‘1) = (𝐹‘1) → 1 = 1) ∧ ((𝐹‘1) = (𝐹‘2) → 1 = 2))))
198186, 197mpbird 246 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦))
199 pm3.22 464 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
2001993adant2 1073 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
201200adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
202201ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
203 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → {𝐶, 𝐴} = {𝐴, 𝐵}))
20433, 202, 203syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → {𝐶, 𝐴} = {𝐴, 𝐵}))
205 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐶𝑉𝐴𝑉) ∧ (𝐴𝑉𝐵𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} ↔ ((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴))))
206205ancoms 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} ↔ ((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴))))
207 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 2 = 0))
20845, 207sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 2 = 0))
2092083ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 2 = 0))
210209com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
211210adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐴𝐴 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
212 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐵 = 𝐶 → (𝐵 = 𝐶 → 2 = 0))
21383, 212sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐵𝐶 → (𝐵 = 𝐶 → 2 = 0))
2142133ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐵 = 𝐶 → 2 = 0))
215214com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐵 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
216215eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐶 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
217216adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐵𝐴 = 𝐴) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
218211, 217jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐴𝐴 = 𝐵) ∨ (𝐶 = 𝐵𝐴 = 𝐴)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0))
219206, 218syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐴, 𝐵} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 0)))
220219com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0)))
221220adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0)))
222221imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0))
223222adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} = {𝐴, 𝐵} → 2 = 0))
224204, 223syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0))
225224adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0))
226 eqeq12 2623 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘2) = (𝐸‘{𝐶, 𝐴}) ∧ (𝐹‘0) = (𝐸‘{𝐴, 𝐵})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
227226ancoms 468 . . . . . . . . . . . . . . . . . 18 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
2282273adant2 1073 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘0) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵})))
229228imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0)))
230229adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐴, 𝐵}) → 2 = 0)))
231225, 230mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘0) → 2 = 0))
232 pm3.22 464 . . . . . . . . . . . . . . . . . . . . 21 (({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
2332323adant1 1072 . . . . . . . . . . . . . . . . . . . 20 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
234233adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
235234ad3antlr 763 . . . . . . . . . . . . . . . . . 18 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))
236 f1ocnvfvrneq 6441 . . . . . . . . . . . . . . . . . 18 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → {𝐶, 𝐴} = {𝐵, 𝐶}))
23733, 235, 236syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → {𝐶, 𝐴} = {𝐵, 𝐶}))
238 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → (𝐶𝑉𝐴𝑉))
239 preq12bg 4326 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐶𝑉𝐴𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ ((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵))))
240238, 42, 239syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} ↔ ((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵))))
241 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝐶 = 𝐴 → (𝐶 = 𝐴 → 2 = 1))
24251, 241sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐶𝐴 → (𝐶 = 𝐴 → 2 = 1))
2432423ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐶 = 𝐴 → 2 = 1))
244243com12 32 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = 𝐴 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
245244eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐶 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
246245adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐵𝐴 = 𝐶) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
247 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝐴 = 𝐵 → (𝐴 = 𝐵 → 2 = 1))
24845, 247sylbi 206 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴𝐵 → (𝐴 = 𝐵 → 2 = 1))
2492483ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝐴 = 𝐵 → 2 = 1))
250249com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 = 𝐵 → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
251250adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 = 𝐶𝐴 = 𝐵) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
252246, 251jaoi 393 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 = 𝐵𝐴 = 𝐶) ∨ (𝐶 = 𝐶𝐴 = 𝐵)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1))
253240, 252syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ({𝐶, 𝐴} = {𝐵, 𝐶} → ((𝐴𝐵𝐵𝐶𝐶𝐴) → 2 = 1)))
254253com23 84 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1)))
255254adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ((𝐴𝐵𝐵𝐶𝐶𝐴) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1)))
256255imp 444 . . . . . . . . . . . . . . . . . 18 (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1))
257256adantr 480 . . . . . . . . . . . . . . . . 17 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ({𝐶, 𝐴} = {𝐵, 𝐶} → 2 = 1))
258237, 257syld 46 . . . . . . . . . . . . . . . 16 ((((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1))
259258adantl 481 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1))
260 eqeq12 2623 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘2) = (𝐸‘{𝐶, 𝐴}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
261260ancoms 468 . . . . . . . . . . . . . . . . . 18 (((𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
2622613adant1 1072 . . . . . . . . . . . . . . . . 17 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → ((𝐹‘2) = (𝐹‘1) ↔ (𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶})))
263262imbi1d 330 . . . . . . . . . . . . . . . 16 (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) → (((𝐹‘2) = (𝐹‘1) → 2 = 1) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1)))
264263adantr 480 . . . . . . . . . . . . . . 15 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘1) → 2 = 1) ↔ ((𝐸‘{𝐶, 𝐴}) = (𝐸‘{𝐵, 𝐶}) → 2 = 1)))
265259, 264mpbird 246 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘1) → 2 = 1))
266 eqidd 2611 . . . . . . . . . . . . . . 15 ((𝐹‘2) = (𝐹‘2) → 2 = 2)
267266a1i 11 . . . . . . . . . . . . . 14 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → ((𝐹‘2) = (𝐹‘2) → 2 = 2))
268231, 265, 2673jca 1235 . . . . . . . . . . . . 13 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
269268adantl 481 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
270104eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 0 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘0)))
271 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 0 → (2 = 𝑦 ↔ 2 = 0))
272270, 271imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 0 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘0) → 2 = 0)))
273108eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 1 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘1)))
274 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 1 → (2 = 𝑦 ↔ 2 = 1))
275273, 274imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 1 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘1) → 2 = 1)))
276112eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑦 = 2 → ((𝐹‘2) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹‘2)))
277 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 2 → (2 = 𝑦 ↔ 2 = 2))
278276, 277imbi12d 333 . . . . . . . . . . . . . 14 (𝑦 = 2 → (((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ ((𝐹‘2) = (𝐹‘2) → 2 = 2)))
279272, 275, 278raltpg 4183 . . . . . . . . . . . . 13 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2))))
280279adantr 480 . . . . . . . . . . . 12 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦) ↔ (((𝐹‘2) = (𝐹‘0) → 2 = 0) ∧ ((𝐹‘2) = (𝐹‘1) → 2 = 1) ∧ ((𝐹‘2) = (𝐹‘2) → 2 = 2))))
281269, 280mpbird 246 . . . . . . . . . . 11 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦))
282118, 198, 2813jca 1235 . . . . . . . . . 10 (((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) ∧ (((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
28329, 282mpan 702 . . . . . . . . 9 ((((𝐹‘0) = (𝐸‘{𝐴, 𝐵}) ∧ (𝐹‘1) = (𝐸‘{𝐵, 𝐶}) ∧ (𝐹‘2) = (𝐸‘{𝐶, 𝐴})) ∧ (((((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐴𝑉)) ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) ∧ (𝐴𝐵𝐵𝐶𝐶𝐴)) ∧ 𝐸:dom 𝐸1-1-onto→ran 𝐸)) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
28425, 283syl 17 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
285 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
286285eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = 0 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘0) = (𝐹𝑦)))
287 eqeq1 2614 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑥 = 𝑦 ↔ 0 = 𝑦))
288286, 287imbi12d 333 . . . . . . . . . . 11 (𝑥 = 0 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
289288ralbidv 2969 . . . . . . . . . 10 (𝑥 = 0 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦)))
290 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 1 → (𝐹𝑥) = (𝐹‘1))
291290eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = 1 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘1) = (𝐹𝑦)))
292 eqeq1 2614 . . . . . . . . . . . 12 (𝑥 = 1 → (𝑥 = 𝑦 ↔ 1 = 𝑦))
293291, 292imbi12d 333 . . . . . . . . . . 11 (𝑥 = 1 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
294293ralbidv 2969 . . . . . . . . . 10 (𝑥 = 1 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦)))
295 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = 2 → (𝐹𝑥) = (𝐹‘2))
296295eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = 2 → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘2) = (𝐹𝑦)))
297 eqeq1 2614 . . . . . . . . . . . 12 (𝑥 = 2 → (𝑥 = 𝑦 ↔ 2 = 𝑦))
298296, 297imbi12d 333 . . . . . . . . . . 11 (𝑥 = 2 → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
299298ralbidv 2969 . . . . . . . . . 10 (𝑥 = 2 → (∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
300289, 294, 299raltpg 4183 . . . . . . . . 9 ((0 ∈ V ∧ 1 ∈ V ∧ 2 ∈ ℤ) → (∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦))))
30129, 300ax-mp 5 . . . . . . . 8 (∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (∀𝑦 ∈ {0, 1, 2} ((𝐹‘0) = (𝐹𝑦) → 0 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘1) = (𝐹𝑦) → 1 = 𝑦) ∧ ∀𝑦 ∈ {0, 1, 2} ((𝐹‘2) = (𝐹𝑦) → 2 = 𝑦)))
302284, 301sylibr 223 . . . . . . 7 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
303 fzo0to3tp 12421 . . . . . . . . 9 (0..^3) = {0, 1, 2}
304303a1i 11 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (0..^3) = {0, 1, 2})
305304raleqdv 3121 . . . . . . . 8 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
306304, 305raleqbidv 3129 . . . . . . 7 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ {0, 1, 2}∀𝑦 ∈ {0, 1, 2} ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
307302, 306mpbird 246 . . . . . 6 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
308 oveq2 6557 . . . . . . . 8 ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = (0..^3))
309308raleqdv 3121 . . . . . . . 8 ((#‘𝐹) = 3 → (∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
310308, 309raleqbidv 3129 . . . . . . 7 ((#‘𝐹) = 3 → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
311310adantr 480 . . . . . 6 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ (0..^3)∀𝑦 ∈ (0..^3)((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
312307, 311mpbird 246 . . . . 5 (((#‘𝐹) = 3 ∧ (𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
3136, 312mpan 702 . . . 4 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
314 dff13 6416 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
3155, 313, 314sylanbrc 695 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)
316 df-f1 5809 . . 3 (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ↔ (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
317315, 316sylib 207 . 2 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ∧ Fun 𝐹))
318317simprd 478 1 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∪ cun 3538  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  3c3 10948  ℤcz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862 This theorem is referenced by:  constr3trl  26187
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