grimcnv - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > grimcnv		Structured version Visualization version GIF version

Theorem grimcnv 47879

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 2737	. . . . 5 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2737	. . . . 5 ⊢ (Vtx‘𝑇) = (Vtx‘𝑇)
3		eqid 2737	. . . . 5 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2737	. . . . 5 ⊢ (iEdg‘𝑇) = (iEdg‘𝑇)
5	1, 2, 3, 4	grimprop 47869	. . . 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 6860	. . . . 5 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → ^◡𝐹:(Vtx‘𝑇)–1-1-onto→(Vtx‘𝑆))
8	7	ad2antrl 728	. . . 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 3484	. . . . . . . . 9 ⊢ 𝑗 ∈ V
10		cnvexg 7946	. . . . . . . . 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 6860	. . . . . . . . . 10 ⊢ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → ^◡𝑗:dom (iEdg‘𝑇)–1-1-onto→dom (iEdg‘𝑆))
13	12	ad2antrl 728	. . . . . . . . 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 6855	. . . . . . . . . . . . 13 ⊢ (𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) → 𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇))
15	14	ad2antrl 728	. . . . . . . . . . . 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 6970	. . . . . . . . . . . 12 ⊢ ((𝑗:dom (iEdg‘𝑆)–onto→dom (iEdg‘𝑇) ∧ 𝑥 ∈ dom (iEdg‘𝑇)) → ∃𝑦 ∈ dom (iEdg‘𝑆)(𝑗‘𝑦) = 𝑥)
17	15, 16	sylan 580	. . . . . . . . . . 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 6911	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑇)‘(𝑗‘𝑖)) = ((iEdg‘𝑇)‘(𝑗‘𝑦)))
19		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑦 → ((iEdg‘𝑆)‘𝑖) = ((iEdg‘𝑆)‘𝑦))
20	19	imaeq2d 6078	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 = 𝑦 → (𝐹 “ ((iEdg‘𝑆)‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦)))
21	18, 20	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝑦 → (((iEdg‘𝑇)‘(𝑗‘𝑖)) = (𝐹 “ ((iEdg‘𝑆)‘𝑖)) ↔ ((iEdg‘𝑇)‘(𝑗‘𝑦)) = (𝐹 “ ((iEdg‘𝑆)‘𝑦))))
22	21	rspcv 3618	. . . . . . . . . . . . . . . . . . . 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 7296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗:dom (iEdg‘𝑆)–1-1-onto→dom (iEdg‘𝑇) ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → (^◡𝑗‘(𝑗‘𝑦)) = 𝑦)
25	24	ad4ant23 753	. . . . . . . . . . . . . . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐹:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝑇) → 𝐹:(Vtx‘𝑆)–1-1→(Vtx‘𝑇))
28	27	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29081	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆 ∈ UHGraph ∧ 𝑦 ∈ dom (iEdg‘𝑆)) → ((iEdg‘𝑆)‘𝑦) ⊆ (Vtx‘𝑆))
30	29	ad5ant15 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6864	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 685	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . . . 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 6078	. . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑦) = 𝑥 → ((𝑗‘𝑦) ∈ dom (iEdg‘𝑇) ↔ 𝑥 ∈ dom (iEdg‘𝑇)))
48		2fveq3 6911	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑆)‘(^◡𝑗‘(𝑗‘𝑦))) = ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)))
49		fveq2 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗‘𝑦) = 𝑥 → ((iEdg‘𝑇)‘(𝑗‘𝑦)) = ((iEdg‘𝑇)‘𝑥))
50	49	imaeq2d 6078	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗‘𝑦) = 𝑥 → (^◡𝐹 “ ((iEdg‘𝑇)‘(𝑗‘𝑦))) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)))
51	48, 50	eqeq12d 2753	. . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . 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 3146	. . . . . . . . 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 6836	. . . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ (𝑓 = ^◡𝑗 → (𝑓‘𝑥) = (^◡𝑗‘𝑥))
63	62	fveqeq2d 6914	. . . . . . . . . 10 ⊢ (𝑓 = ^◡𝑗 → (((iEdg‘𝑆)‘(𝑓‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥)) ↔ ((iEdg‘𝑆)‘(^◡𝑗‘𝑥)) = (^◡𝐹 “ ((iEdg‘𝑇)‘𝑥))))
64	63	ralbidv 3178	. . . . . . . . 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 632	. . . . . . . 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 3598	. . . . . . 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 1933	. . . . 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 47866	. . . . . . . 8 ⊢ Rel dom GraphIso
71	70	ovrcl 7472	. . . . . . 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 7946	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 GraphIso 𝑇) → ^◡𝐹 ∈ V)
75	2, 1, 4, 3	isgrim 47868	. . . . . 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 1373	. . . . 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 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 ∈ (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 713	. . 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 687	. 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 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ^◡ccnv 5684 dom cdm 5685 “ cima 5688 –1-1→wf1 6558 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 GraphIso cgrim 47861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-uhgr 29075 df-grim 47864

This theorem is referenced by: gricsym 47890

W3C validator