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Theorem isomgreqve 44703
 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 6674 . . . 4 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐵) ∈ V)
21resiexd 6971 . . 3 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)) ∈ V)
3 f1oi 6640 . . . . 5 ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)
4 simprl 771 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐴) = (Vtx‘𝐵))
54f1oeq2d 6599 . . . . 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 261 . . . 4 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵))
7 fvexd 6674 . . . . . . 7 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐵) ∈ V)
87dmexd 7616 . . . . . 6 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐵) ∈ V)
98resiexd 6971 . . . . 5 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)) ∈ V)
10 f1oi 6640 . . . . . . 7 ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)
11 simprr 773 . . . . . . . . 9 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐴) = (iEdg‘𝐵))
1211dmeqd 5746 . . . . . . . 8 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
1312f1oeq2d 6599 . . . . . . 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 261 . . . . . 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 2759 . . . . . . . . . . . 12 (Vtx‘𝐴) = (Vtx‘𝐴)
16 eqid 2759 . . . . . . . . . . . 12 (iEdg‘𝐴) = (iEdg‘𝐴)
1715, 16uhgrss 26949 . . . . . . . . . . 11 ((𝐴 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
1817ad4ant14 752 . . . . . . . . . 10 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
19 sseq2 3919 . . . . . . . . . . . . 13 ((Vtx‘𝐴) = (Vtx‘𝐵) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2019adantr 485 . . . . . . . . . . . 12 (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2120adantl 486 . . . . . . . . . . 11 (((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
2221adantr 485 . . . . . . . . . 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 235 . . . . . . . . 9 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵))
24 resiima 5917 . . . . . . . . 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 6927 . . . . . . . . . 10 (𝑖 ∈ dom (iEdg‘𝐴) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
2726adantl 486 . . . . . . . . 9 ((((𝐴 ∈ UHGraph ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
2827fveq2d 6663 . . . . . . . 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 5744 . . . . . . . . . . . . . 14 ((iEdg‘𝐴) = (iEdg‘𝐵) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
3130reseq2d 5824 . . . . . . . . . . . . 13 ((iEdg‘𝐴) = (iEdg‘𝐵) → ( I ↾ dom (iEdg‘𝐴)) = ( I ↾ dom (iEdg‘𝐵)))
3231fveq1d 6661 . . . . . . . . . . . 12 ((iEdg‘𝐴) = (iEdg‘𝐵) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
3329, 32fveq12d 6666 . . . . . . . . . . 11 ((iEdg‘𝐴) = (iEdg‘𝐵) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3433adantl 486 . . . . . . . . . 10 (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
3534adantl 486 . . . . . . . . 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 485 . . . . . . . 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 2800 . . . . . . 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 3114 . . . . . 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 516 . . . . 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 6591 . . . . . 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 6658 . . . . . . . . 9 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
4241fveq2d 6663 . . . . . . . 8 (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
4342eqeq2d 2770 . . . . . . 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 3127 . . . . . 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 634 . . . . 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 3518 . . . 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 516 . . 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 6591 . . . 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 5897 . . . . . . . 8 (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)))
5049eqeq1d 2761 . . . . . . 7 (𝑓 = ( I ↾ (Vtx‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖)) ↔ (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔𝑖))))
5150ralbidv 3127 . . . . . 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 632 . . . . 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 1923 . . . 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 634 . . 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 3518 . 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 2759 . . . 4 (Vtx‘𝐵) = (Vtx‘𝐵)
57 eqid 2759 . . . 4 (iEdg‘𝐵) = (iEdg‘𝐵)
5815, 56, 16, 57isomgr 44701 . . 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 485 . 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 260 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 209   ∧ wa 400   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ⊆ wss 3859   class class class wbr 5033   I cid 5430  dom cdm 5525   ↾ cres 5527   “ cima 5528  –1-1-onto→wf1o 6335  ‘cfv 6336  Vtxcvtx 26881  iEdgciedg 26882  UHGraphcuhgr 26941   IsomGr cisomgr 44697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-uhgr 26943  df-isomgr 44699 This theorem is referenced by:  isomgrref  44713  strisomgrop  44718
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