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Theorem 3vfriswmgralem 26531
 Description: Lemma for 3vfriswmgra 26532. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
3vfriswmgralem (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
Distinct variable groups:   𝑤,𝐴   𝑤,𝐵   𝑤,𝐶   𝑤,𝐸   𝑤,𝑋   𝑤,𝑌

Proof of Theorem 3vfriswmgralem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . 7 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → {𝐴, 𝐵} ∈ ran 𝐸)
21olcd 407 . . . . . 6 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸 ∨ {𝐴, 𝐵} ∈ ran 𝐸))
3 preq2 4213 . . . . . . . . . 10 (𝑤 = 𝐴 → {𝐴, 𝑤} = {𝐴, 𝐴})
43eleq1d 2672 . . . . . . . . 9 (𝑤 = 𝐴 → ({𝐴, 𝑤} ∈ ran 𝐸 ↔ {𝐴, 𝐴} ∈ ran 𝐸))
5 preq2 4213 . . . . . . . . . 10 (𝑤 = 𝐵 → {𝐴, 𝑤} = {𝐴, 𝐵})
65eleq1d 2672 . . . . . . . . 9 (𝑤 = 𝐵 → ({𝐴, 𝑤} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
74, 6rexprg 4182 . . . . . . . 8 ((𝐴𝑋𝐵𝑌) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸 ↔ ({𝐴, 𝐴} ∈ ran 𝐸 ∨ {𝐴, 𝐵} ∈ ran 𝐸)))
873ad2ant1 1075 . . . . . . 7 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸 ↔ ({𝐴, 𝐴} ∈ ran 𝐸 ∨ {𝐴, 𝐵} ∈ ran 𝐸)))
98adantr 480 . . . . . 6 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸 ↔ ({𝐴, 𝐴} ∈ ran 𝐸 ∨ {𝐴, 𝐵} ∈ ran 𝐸)))
102, 9mpbird 246 . . . . 5 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
11 df-rex 2902 . . . . 5 (∃𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸 ↔ ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸))
1210, 11sylib 207 . . . 4 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸))
13 vex 3176 . . . . . . . . 9 𝑤 ∈ V
1413elpr 4146 . . . . . . . 8 (𝑤 ∈ {𝐴, 𝐵} ↔ (𝑤 = 𝐴𝑤 = 𝐵))
15 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
1615elpr 4146 . . . . . . . . . . 11 (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴𝑦 = 𝐵))
17 eqidd 2611 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐴)
1817a1i 11 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐴))
19182a1i 12 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐴))))
20 preq2 4213 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → {𝐴, 𝑦} = {𝐴, 𝐴})
2120eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ ran 𝐸 ↔ {𝐴, 𝐴} ∈ ran 𝐸))
22 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
2322imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐴)))
2423imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐴))))
2519, 21, 243imtr4d 282 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦))))
26 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . . . . 22 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴𝐴)
27 df-ne 2782 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
28 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐴 = 𝐴
2928pm2.24i 145 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴 = 𝐴𝐴 = 𝐵)
3027, 29sylbi 206 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴𝐴𝐴 = 𝐵)
3126, 30syl 17 . . . . . . . . . . . . . . . . . . . . 21 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝐵)
3231ex 449 . . . . . . . . . . . . . . . . . . . 20 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸𝐴 = 𝐵))
33323ad2ant3 1077 . . . . . . . . . . . . . . . . . . 19 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸𝐴 = 𝐵))
3433adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸𝐴 = 𝐵))
3534com12 32 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐵))
36352a1i 12 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐵))))
37 preq2 4213 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
3837eleq1d 2672 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ ran 𝐸 ↔ {𝐴, 𝐵} ∈ ran 𝐸))
39 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
4039imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐵)))
4140imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → (({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝐵))))
4236, 38, 413imtr4d 282 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦))))
4325, 42jaoi 393 . . . . . . . . . . . . . 14 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦))))
44 eqeq1 2614 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝐴 → (𝑤 = 𝑦𝐴 = 𝑦))
4544imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑤 = 𝐴 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦)))
464, 45imbi12d 333 . . . . . . . . . . . . . . 15 (𝑤 = 𝐴 → (({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦))))
4746imbi2d 329 . . . . . . . . . . . . . 14 (𝑤 = 𝐴 → (({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐴 = 𝑦)))))
4843, 47syl5ibr 235 . . . . . . . . . . . . 13 (𝑤 = 𝐴 → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)))))
4928pm2.24i 145 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐴 = 𝐴𝐵 = 𝐴)
5027, 49sylbi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴𝐴𝐵 = 𝐴)
5126, 50syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐵 = 𝐴)
5251ex 449 . . . . . . . . . . . . . . . . . . . . 21 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸𝐵 = 𝐴))
53523ad2ant3 1077 . . . . . . . . . . . . . . . . . . . 20 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸𝐵 = 𝐴))
5453adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸𝐵 = 𝐴))
5554com12 32 . . . . . . . . . . . . . . . . . 18 ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴))
5655a1d 25 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴)))
5756a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴))))
58 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐴 → (𝐵 = 𝑦𝐵 = 𝐴))
5958imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴)))
6059imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴))))
6157, 21, 603imtr4d 282 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦))))
62 eqidd 2611 . . . . . . . . . . . . . . . . . 18 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐵)
6362a1i 11 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐵))
64632a1i 12 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → ({𝐴, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐵))))
65 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝐵 → (𝐵 = 𝑦𝐵 = 𝐵))
6665imbi2d 329 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐵)))
6766imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐵 → (({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐵))))
6864, 38, 673imtr4d 282 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦))))
6961, 68jaoi 393 . . . . . . . . . . . . . 14 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦))))
70 eqeq1 2614 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝐵 → (𝑤 = 𝑦𝐵 = 𝑦))
7170imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑤 = 𝐵 → (((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦)))
726, 71imbi12d 333 . . . . . . . . . . . . . . 15 (𝑤 = 𝐵 → (({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦))))
7372imbi2d 329 . . . . . . . . . . . . . 14 (𝑤 = 𝐵 → (({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))) ↔ ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝑦)))))
7469, 73syl5ibr 235 . . . . . . . . . . . . 13 (𝑤 = 𝐵 → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)))))
7548, 74jaoi 393 . . . . . . . . . . . 12 ((𝑤 = 𝐴𝑤 = 𝐵) → ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)))))
7675com3l 87 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵) → ({𝐴, 𝑦} ∈ ran 𝐸 → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)))))
7716, 76sylbi 206 . . . . . . . . . 10 (𝑦 ∈ {𝐴, 𝐵} → ({𝐴, 𝑦} ∈ ran 𝐸 → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦)))))
7877imp 444 . . . . . . . . 9 ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸) → ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))))
7978com3l 87 . . . . . . . 8 ((𝑤 = 𝐴𝑤 = 𝐵) → ({𝐴, 𝑤} ∈ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸) → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))))
8014, 79sylbi 206 . . . . . . 7 (𝑤 ∈ {𝐴, 𝐵} → ({𝐴, 𝑤} ∈ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸) → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))))
8180imp31 447 . . . . . 6 (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸)) → ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → 𝑤 = 𝑦))
8281com12 32 . . . . 5 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸)) → 𝑤 = 𝑦))
8382alrimivv 1843 . . . 4 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∀𝑤𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸)) → 𝑤 = 𝑦))
84 eleq1 2676 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ {𝐴, 𝐵} ↔ 𝑦 ∈ {𝐴, 𝐵}))
85 preq2 4213 . . . . . . 7 (𝑤 = 𝑦 → {𝐴, 𝑤} = {𝐴, 𝑦})
8685eleq1d 2672 . . . . . 6 (𝑤 = 𝑦 → ({𝐴, 𝑤} ∈ ran 𝐸 ↔ {𝐴, 𝑦} ∈ ran 𝐸))
8784, 86anbi12d 743 . . . . 5 (𝑤 = 𝑦 → ((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸)))
8887eu4 2506 . . . 4 (∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ↔ (∃𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ∧ ∀𝑤𝑦(((𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑦} ∈ ran 𝐸)) → 𝑤 = 𝑦)))
8912, 83, 88sylanbrc 695 . . 3 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸))
90 df-reu 2903 . . 3 (∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸 ↔ ∃!𝑤(𝑤 ∈ {𝐴, 𝐵} ∧ {𝐴, 𝑤} ∈ ran 𝐸))
9189, 90sylibr 223 . 2 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸)
9291ex 449 1 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → ∃!𝑤 ∈ {𝐴, 𝐵} {𝐴, 𝑤} ∈ ran 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∃wrex 2897  ∃!wreu 2898  {cpr 4127  {ctp 4129   class class class wbr 4583  ran crn 5039   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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862 This theorem is referenced by:  3vfriswmgra  26532
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