isomgrsym - Mathbox for Alexander van der Vekens

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Theorem isomgrsym 44075

Description: The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)

**Assertion**
Ref	Expression
isomgrsym	⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 → 𝐵 IsomGr 𝐴))

Proof of Theorem isomgrsym Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2820	. . 3 ⊢ (Vtx‘𝐴) = (Vtx‘𝐴)
2		eqid 2820	. . 3 ⊢ (Vtx‘𝐵) = (Vtx‘𝐵)
3		eqid 2820	. . 3 ⊢ (iEdg‘𝐴) = (iEdg‘𝐴)
4		eqid 2820	. . 3 ⊢ (iEdg‘𝐵) = (iEdg‘𝐵)
5	1, 2, 3, 4	isomgr 44062	. 2 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (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)‘(𝑔‘𝑖))))))
6		vex 3494	. . . . . . . 8 ⊢ 𝑓 ∈ V
7	6	cnvex 7623	. . . . . . 7 ⊢ ^◡𝑓 ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → ^◡𝑓 ∈ V)
9		f1ocnv 6620	. . . . . . . . 9 ⊢ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → ^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴))
10	9	adantr 483	. . . . . . . 8 ⊢ ((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)‘(𝑔‘𝑖)))) → ^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴))
11	10	adantl 484	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → ^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴))
12		vex 3494	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
13	12	cnvex 7623	. . . . . . . . . . . . 13 ⊢ ^◡𝑔 ∈ V
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → ^◡𝑔 ∈ V)
15		f1ocnv 6620	. . . . . . . . . . . . . . 15 ⊢ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → ^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴))
16	15	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → ^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴))
17	16	3ad2ant2 1129	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → ^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴))
18		f1ocnvdm 7034	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (^◡𝑔‘𝑗) ∈ dom (iEdg‘𝐴))
19	18	3ad2antl2 1181	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (^◡𝑔‘𝑗) ∈ dom (iEdg‘𝐴))
20		fveq2 6663	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = (^◡𝑔‘𝑗) → ((iEdg‘𝐴)‘𝑖) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
21	20	imaeq2d 5922	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = (^◡𝑔‘𝑗) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))))
22		2fveq3 6668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = (^◡𝑔‘𝑗) → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗))))
23	21, 22	eqeq12d 2836	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = (^◡𝑔‘𝑗) → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗)))))
24	23	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) ∧ 𝑖 = (^◡𝑔‘𝑗)) → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗)))))
25	19, 24	rspcdv 3612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗)))))
26		f1ocnvfv2 7027	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (𝑔‘(^◡𝑔‘𝑗)) = 𝑗)
27	26	3ad2antl2 1181	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (𝑔‘(^◡𝑔‘𝑗)) = 𝑗)
28	27	fveq2d 6667	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗))
29	28	eqeq2d 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗))) ↔ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗)))
30		f1of1 6607	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → 𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵))
31	30	3ad2ant3 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → 𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵))
32	31	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → 𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵))
33		simpl1l 1219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → 𝐴 ∈ UHGraph)
34	1, 3	uhgrss 26847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐴 ∈ UHGraph ∧ (^◡𝑔‘𝑗) ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) ⊆ (Vtx‘𝐴))
35	33, 19, 34	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) ⊆ (Vtx‘𝐴))
36	32, 35	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵) ∧ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) ⊆ (Vtx‘𝐴)))
37	36	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) ∧ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗)) → (𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵) ∧ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) ⊆ (Vtx‘𝐴)))
38		f1imacnv 6624	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓:(Vtx‘𝐴)–1-1→(Vtx‘𝐵) ∧ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) ⊆ (Vtx‘𝐴)) → (^◡𝑓 “ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) ∧ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗)) → (^◡𝑓 “ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
40		imaeq2 5918	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗) → (^◡𝑓 “ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))
41	40	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) ∧ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗)) → (^◡𝑓 “ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))
42	39, 41	eqtr3d 2857	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) ∧ (𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))
43	42	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘𝑗) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗))))
44	29, 43	sylbid 242	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))) = ((iEdg‘𝐵)‘(𝑔‘(^◡𝑔‘𝑗))) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗))))
45	25, 44	syld 47	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗))))
46	45	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → (𝑗 ∈ dom (iEdg‘𝐵) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))))
47	46	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ 𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → (𝑗 ∈ dom (iEdg‘𝐵) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))))
48	47	3exp 1114	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → (𝑗 ∈ dom (iEdg‘𝐵) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))))))
49	48	com34 91	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (𝑗 ∈ dom (iEdg‘𝐵) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))))))
50	49	impd 413	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → (𝑗 ∈ dom (iEdg‘𝐵) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗))))))
51	50	3imp1 1342	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))
52	51	eqcomd 2826	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) ∧ 𝑗 ∈ dom (iEdg‘𝐵)) → (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
53	52	ralrimiva 3181	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
54	17, 53	jca 514	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → (^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))))
55		f1oeq1 6597	. . . . . . . . . . . . 13 ⊢ (ℎ = ^◡𝑔 → (ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ↔ ^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴)))
56		fveq1 6662	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = ^◡𝑔 → (ℎ‘𝑗) = (^◡𝑔‘𝑗))
57	56	fveq2d 6667	. . . . . . . . . . . . . . 15 ⊢ (ℎ = ^◡𝑔 → ((iEdg‘𝐴)‘(ℎ‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))
58	57	eqeq2d 2831	. . . . . . . . . . . . . 14 ⊢ (ℎ = ^◡𝑔 → ((^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)) ↔ (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))))
59	58	ralbidv 3196	. . . . . . . . . . . . 13 ⊢ (ℎ = ^◡𝑔 → (∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗))))
60	55, 59	anbi12d 632	. . . . . . . . . . . 12 ⊢ (ℎ = ^◡𝑔 → ((ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))) ↔ (^◡𝑔:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(^◡𝑔‘𝑗)))))
61	14, 54, 60	spcedv 3596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵)) → ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))
62	61	3exp 1114	. . . . . . . . . 10 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → ((𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
63	62	exlimdv 1933	. . . . . . . . 9 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))) → (𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) → ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
64	63	com23 86	. . . . . . . 8 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (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)‘(𝑔‘𝑖))) → ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
65	64	imp32 421	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))
66	11, 65	jca 514	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → (^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)))))
67		f1oeq1 6597	. . . . . . 7 ⊢ (𝑒 = ^◡𝑓 → (𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ↔ ^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴)))
68		imaeq1 5917	. . . . . . . . . . 11 ⊢ (𝑒 = ^◡𝑓 → (𝑒 “ ((iEdg‘𝐵)‘𝑗)) = (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)))
69	68	eqeq1d 2822	. . . . . . . . . 10 ⊢ (𝑒 = ^◡𝑓 → ((𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)) ↔ (^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))
70	69	ralbidv 3196	. . . . . . . . 9 ⊢ (𝑒 = ^◡𝑓 → (∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)) ↔ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))
71	70	anbi2d 630	. . . . . . . 8 ⊢ (𝑒 = ^◡𝑓 → ((ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))) ↔ (ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)))))
72	71	exbidv 1921	. . . . . . 7 ⊢ (𝑒 = ^◡𝑓 → (∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))) ↔ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)))))
73	67, 72	anbi12d 632	. . . . . 6 ⊢ (𝑒 = ^◡𝑓 → ((𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)))) ↔ (^◡𝑓:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(^◡𝑓 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
74	8, 66, 73	spcedv 3596	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → ∃𝑒(𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗)))))
75	2, 1, 4, 3	isomgr 44062	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝐴 ∈ UHGraph) → (𝐵 IsomGr 𝐴 ↔ ∃𝑒(𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
76	75	ancoms 461	. . . . . 6 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐵 IsomGr 𝐴 ↔ ∃𝑒(𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
77	76	adantr 483	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → (𝐵 IsomGr 𝐴 ↔ ∃𝑒(𝑒:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐴) ∧ ∃ℎ(ℎ:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐴) ∧ ∀𝑗 ∈ dom (iEdg‘𝐵)(𝑒 “ ((iEdg‘𝐵)‘𝑗)) = ((iEdg‘𝐴)‘(ℎ‘𝑗))))))
78	74, 77	mpbird 259	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ (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)‘(𝑔‘𝑖))))) → 𝐵 IsomGr 𝐴)
79	78	ex 415	. . 3 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → ((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)‘(𝑔‘𝑖)))) → 𝐵 IsomGr 𝐴))
80	79	exlimdv 1933	. 2 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (∃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)‘(𝑔‘𝑖)))) → 𝐵 IsomGr 𝐴))
81	5, 80	sylbid 242	1 ⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 → 𝐵 IsomGr 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ∀wral 3137 Vcvv 3491 ⊆ wss 3929 class class class wbr 5059 ^◡ccnv 5547 dom cdm 5548 “ cima 5551 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 Vtxcvtx 26779 iEdgciedg 26780 UHGraphcuhgr 26839 IsomGr cisomgr 44058

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-uhgr 26841 df-isomgr 44060

This theorem is referenced by: isomgrsymb 44076

W3C validator