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Theorem usgrares1 25939
 Description: Restricting an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
Hypothesis
Ref Expression
usgrafis.f 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
Assertion
Ref Expression
usgrares1 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
Distinct variable groups:   𝑥,𝐸   𝑥,𝑁
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem usgrares1
Dummy variables 𝑦 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgraf1 25889 . . . . . 6 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
21adantr 480 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → 𝐸:dom 𝐸1-1→ran 𝐸)
3 ssrab2 3650 . . . . 5 {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸
4 f1ssres 6021 . . . . 5 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸) → (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}):{𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}–1-1→ran 𝐸)
52, 3, 4sylancl 693 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}):{𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}–1-1→ran 𝐸)
6 usgrafis.f . . . . . 6 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
76a1i 11 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}))
86dmeqi 5247 . . . . . 6 dom 𝐹 = dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
93a1i 11 . . . . . . 7 ((𝑉 USGrph 𝐸𝑁𝑉) → {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸)
10 ssdmres 5340 . . . . . . 7 ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸 ↔ dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
119, 10sylib 207 . . . . . 6 ((𝑉 USGrph 𝐸𝑁𝑉) → dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
128, 11syl5eq 2656 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → dom 𝐹 = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
13 eqidd 2611 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → ran 𝐸 = ran 𝐸)
147, 12, 13f1eq123d 6044 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝐹:dom 𝐹1-1→ran 𝐸 ↔ (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}):{𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}–1-1→ran 𝐸))
155, 14mpbird 246 . . 3 ((𝑉 USGrph 𝐸𝑁𝑉) → 𝐹:dom 𝐹1-1→ran 𝐸)
166rneqi 5273 . . . . 5 ran 𝐹 = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
17 df-ima 5051 . . . . 5 (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = ran (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
1816, 17eqtr4i 2635 . . . 4 ran 𝐹 = (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
19 eqidd 2611 . . . . . . . . 9 (𝑥 = 𝑦𝑁 = 𝑁)
20 fveq2 6103 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐸𝑥) = (𝐸𝑦))
2119, 20neleq12d 2887 . . . . . . . 8 (𝑥 = 𝑦 → (𝑁 ∉ (𝐸𝑥) ↔ 𝑁 ∉ (𝐸𝑦)))
2221elrab 3331 . . . . . . 7 (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ↔ (𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)))
23 usgraf0 25877 . . . . . . . . 9 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
24 f1f 6014 . . . . . . . . . . . . . . 15 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → 𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
25 ffvelrn 6265 . . . . . . . . . . . . . . . . 17 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑦 ∈ dom 𝐸) → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2})
26 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝐸𝑦) → (#‘𝑒) = (#‘(𝐸𝑦)))
2726eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝐸𝑦) → ((#‘𝑒) = 2 ↔ (#‘(𝐸𝑦)) = 2))
2827elrab 3331 . . . . . . . . . . . . . . . . . 18 ((𝐸𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ↔ ((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2))
29 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∉ (𝐸𝑦) ↔ ¬ 𝑁 ∈ (𝐸𝑦))
3029biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∉ (𝐸𝑦) → ¬ 𝑁 ∈ (𝐸𝑦))
3130ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → ¬ 𝑁 ∈ (𝐸𝑦))
32 disjsn 4192 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸𝑦) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (𝐸𝑦))
3331, 32sylibr 223 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → ((𝐸𝑦) ∩ {𝑁}) = ∅)
34 elpwi 4117 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐸𝑦) ∈ 𝒫 𝑉 → (𝐸𝑦) ⊆ 𝑉)
3534ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → (𝐸𝑦) ⊆ 𝑉)
36 reldisj 3972 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸𝑦) ⊆ 𝑉 → (((𝐸𝑦) ∩ {𝑁}) = ∅ ↔ (𝐸𝑦) ⊆ (𝑉 ∖ {𝑁})))
3735, 36syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → (((𝐸𝑦) ∩ {𝑁}) = ∅ ↔ (𝐸𝑦) ⊆ (𝑉 ∖ {𝑁})))
3833, 37mpbid 221 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → (𝐸𝑦) ⊆ (𝑉 ∖ {𝑁}))
39 fvex 6113 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸𝑦) ∈ V
4039elpw 4114 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ↔ (𝐸𝑦) ⊆ (𝑉 ∖ {𝑁}))
4138, 40sylibr 223 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → (𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}))
42 simpllr 795 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → (#‘(𝐸𝑦)) = 2)
4341, 42jca 553 . . . . . . . . . . . . . . . . . . 19 (((((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) ∧ 𝑁 ∉ (𝐸𝑦)) ∧ 𝑁𝑉) → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))
4443exp31 628 . . . . . . . . . . . . . . . . . 18 (((𝐸𝑦) ∈ 𝒫 𝑉 ∧ (#‘(𝐸𝑦)) = 2) → (𝑁 ∉ (𝐸𝑦) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))))
4528, 44sylbi 206 . . . . . . . . . . . . . . . . 17 ((𝐸𝑦) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁 ∉ (𝐸𝑦) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))))
4625, 45syl 17 . . . . . . . . . . . . . . . 16 ((𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑦 ∈ dom 𝐸) → (𝑁 ∉ (𝐸𝑦) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))))
4746ex 449 . . . . . . . . . . . . . . 15 (𝐸:dom 𝐸⟶{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑦 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑦) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2)))))
4824, 47syl 17 . . . . . . . . . . . . . 14 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑦 ∈ dom 𝐸 → (𝑁 ∉ (𝐸𝑦) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2)))))
4948impd 446 . . . . . . . . . . . . 13 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → ((𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)) → (𝑁𝑉 → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))))
5049com23 84 . . . . . . . . . . . 12 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁𝑉 → ((𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)) → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))))
5150imp31 447 . . . . . . . . . . 11 (((𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑁𝑉) ∧ (𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦))) → ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))
5227elrab 3331 . . . . . . . . . . 11 ((𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ((𝐸𝑦) ∈ 𝒫 (𝑉 ∖ {𝑁}) ∧ (#‘(𝐸𝑦)) = 2))
5351, 52sylibr 223 . . . . . . . . . 10 (((𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} ∧ 𝑁𝑉) ∧ (𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦))) → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
5453exp31 628 . . . . . . . . 9 (𝐸:dom 𝐸1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2} → (𝑁𝑉 → ((𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)) → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})))
5523, 54syl 17 . . . . . . . 8 (𝑉 USGrph 𝐸 → (𝑁𝑉 → ((𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)) → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})))
5655imp 444 . . . . . . 7 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑦 ∈ dom 𝐸𝑁 ∉ (𝐸𝑦)) → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
5722, 56syl5bi 231 . . . . . 6 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} → (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
5857ralrimiv 2948 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → ∀𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
59 usgrafun 25878 . . . . . . . 8 (𝑉 USGrph 𝐸 → Fun 𝐸)
603a1i 11 . . . . . . . 8 (𝑉 USGrph 𝐸 → {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸)
6159, 60jca 553 . . . . . . 7 (𝑉 USGrph 𝐸 → (Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸))
6261adantr 480 . . . . . 6 ((𝑉 USGrph 𝐸𝑁𝑉) → (Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸))
63 funimass4 6157 . . . . . 6 ((Fun 𝐸 ∧ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ⊆ dom 𝐸) → ((𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ∀𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
6462, 63syl 17 . . . . 5 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2} ↔ ∀𝑦 ∈ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} (𝐸𝑦) ∈ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
6558, 64mpbird 246 . . . 4 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝐸 “ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
6618, 65syl5eqss 3612 . . 3 ((𝑉 USGrph 𝐸𝑁𝑉) → ran 𝐹 ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
67 f1ssr 6020 . . 3 ((𝐹:dom 𝐹1-1→ran 𝐸 ∧ ran 𝐹 ⊆ {𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}) → 𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
6815, 66, 67syl2anc 691 . 2 ((𝑉 USGrph 𝐸𝑁𝑉) → 𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2})
69 usgrav 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
70 difexg 4735 . . . . 5 (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
71 resexg 5362 . . . . . 6 (𝐸 ∈ V → (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) ∈ V)
726, 71syl5eqel 2692 . . . . 5 (𝐸 ∈ V → 𝐹 ∈ V)
73 isusgra0 25876 . . . . 5 (((𝑉 ∖ {𝑁}) ∈ V ∧ 𝐹 ∈ V) → ((𝑉 ∖ {𝑁}) USGrph 𝐹𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
7470, 72, 73syl2an 493 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉 ∖ {𝑁}) USGrph 𝐹𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
7569, 74syl 17 . . 3 (𝑉 USGrph 𝐸 → ((𝑉 ∖ {𝑁}) USGrph 𝐹𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
7675adantr 480 . 2 ((𝑉 USGrph 𝐸𝑁𝑉) → ((𝑉 ∖ {𝑁}) USGrph 𝐹𝐹:dom 𝐹1-1→{𝑒 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑒) = 2}))
7768, 76mpbird 246 1 ((𝑉 USGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) USGrph 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   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-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-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-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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:  usgrafis  25944  cusgrares  26001
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