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Theorem isomgreqve 43989
Description: A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.)
Assertion
Ref Expression
isomgreqve (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵)

Proof of Theorem isomgreqve
Dummy variables 𝑓 𝑔 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6684 . . . 4 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐵) ∈ V)
21resiexd 6978 . . 3 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)) ∈ V)
3 f1oi 6651 . . . . 5 ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)
4 simprl 769 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐴) = (Vtx‘𝐵))
54f1oeq2d 6610 . . . . 5 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ↔ ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)))
63, 5mpbiri 260 . . . 4 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵))
7 fvexd 6684 . . . . . . 7 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐵) ∈ V)
87dmexd 7614 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐵) ∈ V)
98resiexd 6978 . . . . 5 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)) ∈ V)
10 f1oi 6651 . . . . . . 7 ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)
11 simprr 771 . . . . . . . . 9 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐴) = (iEdg‘𝐵))
1211dmeqd 5773 . . . . . . . 8 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
1312f1oeq2d 6610 . . . . . . 7 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ↔ ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)))
1410, 13mpbiri 260 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵))
15 eqid 2821 . . . . . . . . . . . 12 (Vtx‘𝐴) = (Vtx‘𝐴)
16 eqid 2821 . . . . . . . . . . . 12 (iEdg‘𝐴) = (iEdg‘𝐴)
1715, 16uhgrss 26848 . . . . . . . . . . 11 ((𝐴 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
1817ad4ant14 750 . . . . . . . . . 10 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
19 sseq2 3992 . . . . . . . . . . . . 13 ((Vtx‘𝐴) = (Vtx‘𝐵) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2019adantr 483 . . . . . . . . . . . 12 (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2120adantl 484 . . . . . . . . . . 11 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2221adantr 483 . . . . . . . . . 10 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2318, 22mpbid 234 . . . . . . . . 9 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵))
24 resiima 5943 . . . . . . . . 9 (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵) → (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐴)‘𝑖))
2523, 24syl 17 . . . . . . . 8 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐴)‘𝑖))
26 fvresi 6934 . . . . . . . . . 10 (𝑖 ∈ dom (iEdg‘𝐴) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
2726adantl 484 . . . . . . . . 9 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
2827fveq2d 6673 . . . . . . . 8 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐴)‘𝑖))
29 id 22 . . . . . . . . . . . 12 ((iEdg‘𝐴) = (iEdg‘𝐵) → (iEdg‘𝐴) = (iEdg‘𝐵))
30 dmeq 5771 . . . . . . . . . . . . . 14 ((iEdg‘𝐴) = (iEdg‘𝐵) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
3130reseq2d 5852 . . . . . . . . . . . . 13 ((iEdg‘𝐴) = (iEdg‘𝐵) → ( I ↾ dom (iEdg‘𝐴)) = ( I ↾ dom (iEdg‘𝐵)))
3231fveq1d 6671 . . . . . . . . . . . 12 ((iEdg‘𝐴) = (iEdg‘𝐵) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
3329, 32fveq12d 6676 . . . . . . . . . . 11 ((iEdg‘𝐴) = (iEdg‘𝐵) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3433adantl 484 . . . . . . . . . 10 (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3534adantl 484 . . . . . . . . 9 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3635adantr 483 . . . . . . . 8 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3725, 28, 363eqtr2d 2862 . . . . . . 7 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3837ralrimiva 3182 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3914, 38jca 514 . . . . 5 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖))))
40 f1oeq1 6603 . . . . . 6 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ↔ ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵)))
41 fveq1 6668 . . . . . . . . 9 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
4241fveq2d 6673 . . . . . . . 8 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
4342eqeq2d 2832 . . . . . . 7 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)) ↔ (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖))))
4443ralbidv 3197 . . . . . 6 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖))))
4540, 44anbi12d 632 . . . . 5 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))) ↔ (( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))))
469, 39, 45spcedv 3598 . . . 4 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))
476, 46jca 514 . . 3 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)))))
48 f1oeq1 6603 . . . 4 (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ↔ ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)))
49 imaeq1 5923 . . . . . . . 8 (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)))
5049eqeq1d 2823 . . . . . . 7 (𝑓 = ( I ↾ (Vtx‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)) ↔ (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))
5150ralbidv 3197 . . . . . 6 (𝑓 = ( I ↾ (Vtx‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))
5251anbi2d 630 . . . . 5 (𝑓 = ( I ↾ (Vtx‘𝐵)) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))) ↔ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)))))
5352exbidv 1918 . . . 4 (𝑓 = ( I ↾ (Vtx‘𝐵)) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))) ↔ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)))))
5448, 53anbi12d 632 . . 3 (𝑓 = ( I ↾ (Vtx‘𝐵)) → ((𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)))) ↔ (( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))))
552, 47, 54spcedv 3598 . 2 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)))))
56 eqid 2821 . . . 4 (Vtx‘𝐵) = (Vtx‘𝐵)
57 eqid 2821 . . . 4 (iEdg‘𝐵) = (iEdg‘𝐵)
5815, 56, 16, 57isomgr 43987 . . 3 ((𝐴 ∈ UHGraph ∧ 𝐵𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))))
5958adantr 483 . 2 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))))
6055, 59mpbird 259 1 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wral 3138  Vcvv 3494  wss 3935   class class class wbr 5065   I cid 5458  dom cdm 5554  cres 5556  cima 5557  1-1-ontowf1o 6353  cfv 6354  Vtxcvtx 26780  iEdgciedg 26781  UHGraphcuhgr 26840   IsomGr cisomgr 43983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-uhgr 26842  df-isomgr 43985
This theorem is referenced by:  isomgrref  43999  strisomgrop  44004
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