Theorem isomgreqve 44343

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 6660	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐵) ∈ V)
2	1	resiexd 6956	. . 3 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)) ∈ V)
3		f1oi 6627	. . . . 5 ⊢ ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)
4		simprl 770	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐴) = (Vtx‘𝐵))
5	4	f1oeq2d 6586	. . . . 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 261	. . . 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 6660	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐵) ∈ V)
8	7	dmexd 7596	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐵) ∈ V)
9	8	resiexd 6956	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)) ∈ V)
10		f1oi 6627	. . . . . . 7 ⊢ ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)
11		simprr 772	. . . . . . . . 9 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐴) = (iEdg‘𝐵))
12	11	dmeqd 5738	. . . . . . . 8 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
13	12	f1oeq2d 6586	. . . . . . 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 261	. . . . . 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 2798	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐴) = (Vtx‘𝐴)
16		eqid 2798	. . . . . . . . . . . 12 ⊢ (iEdg‘𝐴) = (iEdg‘𝐴)
17	15, 16	uhgrss 26857	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
18	17	ad4ant14 751	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
19		sseq2 3941	. . . . . . . . . . . . 13 ⊢ ((Vtx‘𝐴) = (Vtx‘𝐵) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
20	19	adantr 484	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵)) → (((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴) ↔ ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵)))
21	20	adantl 485	. . . . . . . . . . 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 484	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐵))
24		resiima 5911	. . . . . . . . 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 6912	. . . . . . . . . 10 ⊢ (𝑖 ∈ dom (iEdg‘𝐴) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = 𝑖)
27	26	adantl 485	. . . . . . . . 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 6649	. . . . . . . 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 5736	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
31	30	reseq2d 5818	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → ( I ↾ dom (iEdg‘𝐴)) = ( I ↾ dom (iEdg‘𝐵)))
32	31	fveq1d 6647	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
33	29, 32	fveq12d 6652	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → ((iEdg‘𝐴)‘(( I ↾ dom (iEdg‘𝐴))‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
34	33	adantl 485	. . . . . . . . . 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 485	. . . . . . . . 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 484	. . . . . . . 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 2839	. . . . . . 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 3149	. . . . . 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 515	. . . . 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 6579	. . . . . 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 6644	. . . . . . . . 9 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
42	41	fveq2d 6649	. . . . . . . 8 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
43	42	eqeq2d 2809	. . . . . . 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 3162	. . . . . 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 633	. . . . 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 3547	. . . 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 515	. . 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 6579	. . . 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 5891	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)))
50	49	eqeq1d 2800	. . . . . . 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 3162	. . . . . 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 631	. . . . 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 1922	. . . 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 633	. . 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 3547	. 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 2798	. . . 4 ⊢ (Vtx‘𝐵) = (Vtx‘𝐵)
57		eqid 2798	. . . 4 ⊢ (iEdg‘𝐵) = (iEdg‘𝐵)
58	15, 56, 16, 57	isomgr 44341	. . 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 484	. 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 260	1 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   I cid 5424  dom cdm 5519   ↾ cres 5521   “ cima 5522  –1-1-onto→wf1o 6323  ‘cfv 6324  Vtxcvtx 26789  iEdgciedg 26790  UHGraphcuhgr 26849   IsomGr cisomgr 44337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-uhgr 26851  df-isomgr 44339

This theorem is referenced by:  isomgrref  44353  strisomgrop  44358