grimcnv - Mathbox for Alexander van der Vekens

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Theorem grimcnv 47763

Description: The converse of a graph isomorphism is a graph isomorphism. (Contributed by AV, 1-May-2025.)

**Assertion**
Ref	Expression
grimcnv	⊢ (𝑆 ∈ UHGraph → (𝐹 ∈ (𝑆 GraphIso 𝑇) → ^◡𝐹 ∈ (𝑇 GraphIso 𝑆)))

Proof of Theorem grimcnv Dummy variables 𝑓 𝑗 𝑥 𝑖 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . 5 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2740	. . . . 5 ⊢ (Vtx‘𝑇) = (Vtx‘𝑇)
3		eqid 2740	. . . . 5 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2740	. . . . 5 ⊢ (iEdg‘𝑇) = (iEdg‘𝑇)
5	1, 2, 3, 4	grimprop 47753	. . . 4 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))))
6	5	adantl 481	. . 3 ⊢ ((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) → (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))))
7		f1ocnv 6874	. . . . 5 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → ^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆))
8	7	ad2antrl 727	. . . 4 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))))) → ^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆))
9		vex 3492	. . . . . . . . 9 ⊢ 𝑗 ∈ V
10		cnvexg 7964	. . . . . . . . 9 ⊢ (𝑗 ∈ V → ^◡𝑗 ∈ V)
11	9, 10	mp1i 13	. . . . . . . 8 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → ^◡𝑗 ∈ V)
12		f1ocnv 6874	. . . . . . . . . 10 ⊢ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → ^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆))
13	12	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → ^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆))
14		f1ofo 6869	. . . . . . . . . . . . 13 ⊢ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
15	14	ad2antrl 727	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
16		foelcdmi 6983	. . . . . . . . . . . 12 ⊢ ((𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥)
17	15, 16	sylan 579	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥)
18		2fveq3 6925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑇)‘(𝑗‘𝑖)) = ((iEdg‘𝑇)‘(𝑗‘𝑦)))
19		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑆)‘𝑖) = ((iEdg‘𝑆)‘𝑦))
20	19	imaeq2d 6089	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑦 → (𝐹 “ ((iEdg‘𝑆)‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)))
21	18, 20	eqeq12d 2756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑦 → (((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) ↔ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
22	21	rspcv 3631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ dom (iEdg‘𝑆) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
23	22	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
24		f1ocnvfv1 7312	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (^◡𝑗‘(𝑗‘𝑦)) = 𝑦)
25	24	ad4ant23 752	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → (^◡𝑗‘(𝑗‘𝑦)) = 𝑦)
26	25	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = ((iEdg‘𝑆)‘𝑦))
27		f1of1 6861	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → 𝐹:(Vtx‘𝑆)–1-1→(Vtx‘𝑇))
28	27	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → 𝐹:(Vtx‘𝑆)–1-1→(Vtx‘𝑇))
29	1, 3	uhgrss 29099	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆 ∈ UHGraph ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆))
30	29	ad5ant15 758	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆))
31		f1imacnv 6878	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐹:(Vtx‘𝑆)–1-1→(Vtx‘𝑇) ∧ ((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆)) → (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))) = ((iEdg‘𝑆)‘𝑦))
32	28, 30, 31	syl2an2r 684	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))) = ((iEdg‘𝑆)‘𝑦))
33	32	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) = (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘𝑦) = (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
35	26, 34	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
36	35	adantlr 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
37		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)))
38	37	eqcomd 2746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → (𝐹 “ ((iEdg‘𝑆)‘𝑦)) = ((iEdg‘𝑇)‘(𝑗‘𝑦)))
39	38	imaeq2d 6089	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → (^◡𝐹 “ (𝐹 “ ((iEdg‘𝑆)‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))
40	36, 39	eqtrd 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) ∧ (𝑗‘𝑦) ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))
41	40	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) ∧ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦)))))
42	41	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))))
43	23, 42	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))))
44	43	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (𝑦 ∈ dom (iEdg‘𝑆) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦)))))))
45	44	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ 𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇)) → (∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦)))))))
46	45	impr 454	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))))
47		eleq1 2832	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑦) = 𝑥 → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) ↔ 𝑥 ∈ dom (iEdg‘𝑇)))
48		2fveq3 6925	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)))
49		fveq2 6920	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = ((iEdg‘𝑇)‘𝑥))
50	49	imaeq2d 6089	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗‘𝑦) = 𝑥 → (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))
51	48, 50	eqeq12d 2756	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑦) = 𝑥 → (((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))) ↔ ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
52	47, 51	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗‘𝑦) = 𝑥 → (((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦)))) ↔ (𝑥 ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))))
53	52	imbi2d 340	. . . . . . . . . . . . . . 15 ⊢ ((𝑗‘𝑦) = 𝑥 → ((𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))))) ↔ (𝑦 ∈ dom (iEdg‘𝑆) → (𝑥 ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
54	46, 53	syl5ibcom 245	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → ((𝑗‘𝑦) = 𝑥 → (𝑦 ∈ dom (iEdg‘𝑆) → (𝑥 ∈ dom (iEdg‘𝑇) → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
55	54	com24 95	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → (𝑥 ∈ dom (iEdg‘𝑇) → (𝑦 ∈ dom (iEdg‘𝑆) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
56	55	imp31 417	. . . . . . . . . . . 12 ⊢ ((((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
57	56	rexlimdva 3161	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → (∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
58	17, 57	mpd 15	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))
59	58	ralrimiva 3152	. . . . . . . . 9 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))
60	13, 59	jca 511	. . . . . . . 8 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → (^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
61		f1oeq1 6850	. . . . . . . . 9 ⊢ (𝑓 = ^◡𝑗 → (𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ↔ ^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆)))
62		fveq1 6919	. . . . . . . . . . 11 ⊢ (𝑓 = ^◡𝑗 → (𝑓‘𝑥) = (^◡𝑗‘𝑥))
63	62	fveqeq2d 6928	. . . . . . . . . 10 ⊢ (𝑓 = ^◡𝑗 → (((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)) ↔ ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
64	63	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑓 = ^◡𝑗 → (∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)) ↔ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
65	61, 64	anbi12d 631	. . . . . . . 8 ⊢ (𝑓 = ^◡𝑗 → ((𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))) ↔ (^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))))
66	11, 60, 65	spcedv 3611	. . . . . . 7 ⊢ ((((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) ∧ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)))) → ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
67	66	ex 412	. . . . . 6 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))))
68	67	exlimdv 1932	. . . . 5 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ 𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇)) → (∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))) → ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))))
69	68	impr 454	. . . 4 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))))) → ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
70		grimdmrel 47750	. . . . . . . 8 ⊢ Rel dom GraphIso
71	70	ovrcl 7489	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (𝑆 ∈ V ∧ 𝑇 ∈ V))
72	71	simprd 495	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → 𝑇 ∈ V)
73	71	simpld 494	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → 𝑆 ∈ V)
74		cnvexg 7964	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → ^◡𝐹 ∈ V)
75	2, 1, 4, 3	isgrim 47752	. . . . . 6 ⊢ ((𝑇 ∈ V ∧ 𝑆 ∈ V ∧ ^◡𝐹 ∈ V) → (^◡𝐹 ∈ (𝑇 GraphIso 𝑆) ↔ (^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
76	72, 73, 74, 75	syl3anc 1371	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → (^◡𝐹 ∈ (𝑇 GraphIso 𝑆) ↔ (^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
77	76	ad2antlr 726	. . . 4 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))))) → (^◡𝐹 ∈ (𝑇 GraphIso 𝑆) ↔ (^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆) ∧ ∃𝑓(𝑓:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆) ∧ ∀𝑥 ∈ dom (iEdg‘𝑇)((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))))
78	8, 69, 77	mpbir2and 712	. . 3 ⊢ (((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) ∧ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) ∧ ∃𝑗(𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ ∀𝑖 ∈ dom (iEdg‘𝑆)((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖))))) → ^◡𝐹 ∈ (𝑇 GraphIso 𝑆))
79	6, 78	mpdan 686	. 2 ⊢ ((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇)) → ^◡𝐹 ∈ (𝑇 GraphIso 𝑆))
80	79	ex 412	1 ⊢ (𝑆 ∈ UHGraph → (𝐹 ∈ (𝑆 GraphIso 𝑇) → ^◡𝐹 ∈ (𝑇 GraphIso 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ⊆ wss 3976 ^◡ccnv 5699 dom cdm 5700 “ cima 5703 –1-1→wf1 6570 –onto→wfo 6571 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 Vtxcvtx 29031 iEdgciedg 29032 UHGraphcuhgr 29091 GraphIso cgrim 47745

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-map 8886 df-uhgr 29093 df-grim 47748

This theorem is referenced by: gricsym 47774

W3C validator