Theorem isomgreqve 45277

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 ∧ 𝐵 ∈ 𝑌) ∧ ((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.

Step	Hyp	Ref	Expression
1		fvexd 6789	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐵) ∈ V)
2	1	resiexd 7092	. . 3 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)) ∈ V)
3		f1oi 6754	. . . . 5 ⊢ ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)
4		simprl 768	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐴) = (Vtx‘𝐵))
5	4	f1oeq2d 6712	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)))
6	3, 5	mpbiri 257	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵))
7		fvexd 6789	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐵) ∈ V)
8	7	dmexd 7752	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐵) ∈ V)
9	8	resiexd 7092	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)) ∈ V)
10		f1oi 6754	. . . . . . 7 ⊢ ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)
11		simprr 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐴) = (iEdg‘𝐵))
12	11	dmeqd 5814	. . . . . . . 8 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
13	12	f1oeq2d 6712	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)))
14	10, 13	mpbiri 257	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵))
15		eqid 2738	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐴) = (Vtx‘𝐴)
16		eqid 2738	. . . . . . . . . . . 12 ⊢ (iEdg‘𝐴) = (iEdg‘𝐴)
17	15, 16	uhgrss 27434	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
18	17	ad4ant14 749	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
19		sseq2 3947	. . . . . . . . . . . . 13 ⊢ ((Vtx‘𝐴) = (Vtx‘𝐵) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
20	19	adantr 481	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
21	20	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
22	21	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
23	18, 22	mpbid 231	. . . . . . . . 9 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵))
24		resiima 5984	. . . . . . . . 9 ⊢ (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵) → (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐴)‘𝑖))
25	23, 24	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐴)‘𝑖))
26		fvresi 7045	. . . . . . . . . 10 ⊢ (𝑖 ∈ dom (iEdg‘𝐴) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
27	26	adantl 482	. . . . . . . . 9 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
28	27	fveq2d 6778	. . . . . . . 8 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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 5812	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
31	30	reseq2d 5891	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → ( I ↾ dom (iEdg‘𝐴)) = ( I ↾ dom (iEdg‘𝐵)))
32	31	fveq1d 6776	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
33	29, 32	fveq12d 6781	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
34	33	adantl 482	. . . . . . . . . 10 ⊢ (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
35	34	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
36	35	adantr 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)))
37	25, 28, 36	3eqtr2d 2784	. . . . . . 7 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)))
38	37	ralrimiva 3103	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)))
39	14, 38	jca 512	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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 6704	. . . . . 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 6773	. . . . . . . . 9 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
42	41	fveq2d 6778	. . . . . . . 8 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
43	42	eqeq2d 2749	. . . . . . 7 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖))))
44	43	ralbidv 3112	. . . . . 6 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖))))
45	40, 44	anbi12d 631	. . . . 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)‘(𝑔‘𝑖))) ↔ (( 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)))))
46	9, 39, 45	spcedv 3537	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)‘(𝑔‘𝑖))))
47	6, 46	jca 512	. . 3 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)‘(𝑔‘𝑖)))))
48		f1oeq1 6704	. . . 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 5964	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)))
50	49	eqeq1d 2740	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐵)) → ((𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))
51	50	ralbidv 3112	. . . . . 6 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐵)) → (∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖))))
52	51	anbi2d 629	. . . . 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)‘(𝑔‘𝑖))) ↔ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))))
53	52	exbidv 1924	. . . 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)‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))))
54	48, 53	anbi12d 631	. . 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)‘(𝑔‘𝑖)))) ↔ (( 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)‘(𝑔‘𝑖))))))
55	2, 47, 54	spcedv 3537	. 2 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)‘(𝑔‘𝑖)))))
56		eqid 2738	. . . 4 ⊢ (Vtx‘𝐵) = (Vtx‘𝐵)
57		eqid 2738	. . . 4 ⊢ (iEdg‘𝐵) = (iEdg‘𝐵)
58	15, 56, 16, 57	isomgr 45275	. . 3 ⊢ ((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))))))
59	58	adantr 481	. 2 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((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)‘(𝑔‘𝑖))))))
60	55, 59	mpbird 256	1 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   I cid 5488  dom cdm 5589   ↾ cres 5591   “ cima 5592  –1-1-onto→wf1o 6432  ‘cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  UHGraphcuhgr 27426   IsomGr cisomgr 45271

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-uhgr 27428  df-isomgr 45273

This theorem is referenced by:  isomgrref  45287  strisomgrop  45292