isomgreqve - Mathbox for Alexander van der Vekens

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Theorem isomgreqve 46479

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 6903	. . . 4 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐵) ∈ V)
2	1	resiexd 7214	. . 3 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ (Vtx‘𝐵)) ∈ V)
3		f1oi 6868	. . . . 5 ⊢ ( I ↾ (Vtx‘𝐵)):(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐵)
4		simprl 769	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (Vtx‘𝐴) = (Vtx‘𝐵))
5	4	f1oeq2d 6826	. . . . 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 6903	. . . . . . 7 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐵) ∈ V)
8	7	dmexd 7892	. . . . . 6 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐵) ∈ V)
9	8	resiexd 7214	. . . . 5 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → ( I ↾ dom (iEdg‘𝐵)) ∈ V)
10		f1oi 6868	. . . . . . 7 ⊢ ( I ↾ dom (iEdg‘𝐵)):dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐵)
11		simprr 771	. . . . . . . . 9 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → (iEdg‘𝐴) = (iEdg‘𝐵))
12	11	dmeqd 5903	. . . . . . . 8 ⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
13	12	f1oeq2d 6826	. . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (Vtx‘𝐴) = (Vtx‘𝐴)
16		eqid 2732	. . . . . . . . . . . 12 ⊢ (iEdg‘𝐴) = (iEdg‘𝐴)
17	15, 16	uhgrss 28313	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
18	17	ad4ant14 750	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) ∧ 𝑖 ∈ dom (iEdg‘𝐴)) → ((iEdg‘𝐴)‘𝑖) ⊆ (Vtx‘𝐴))
19		sseq2 4007	. . . . . . . . . . . . 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 6072	. . . . . . . . 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 7167	. . . . . . . . . 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 6892	. . . . . . . 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 5901	. . . . . . . . . . . . . 14 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → dom (iEdg‘𝐴) = dom (iEdg‘𝐵))
31	30	reseq2d 5979	. . . . . . . . . . . . 13 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → ( I ↾ dom (iEdg‘𝐴)) = ( I ↾ dom (iEdg‘𝐵)))
32	31	fveq1d 6890	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐴) = (iEdg‘𝐵) → (( I ↾ dom (iEdg‘𝐴))‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
33	29, 32	fveq12d 6895	. . . . . . . . . . 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 2778	. . . . . . 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 3146	. . . . . 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 6818	. . . . . 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 6887	. . . . . . . . 9 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → (𝑔‘𝑖) = (( I ↾ dom (iEdg‘𝐵))‘𝑖))
42	41	fveq2d 6892	. . . . . . . 8 ⊢ (𝑔 = ( I ↾ dom (iEdg‘𝐵)) → ((iEdg‘𝐵)‘(𝑔‘𝑖)) = ((iEdg‘𝐵)‘(( I ↾ dom (iEdg‘𝐵))‘𝑖)))
43	42	eqeq2d 2743	. . . . . . 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 3177	. . . . . 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 3588	. . . 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 6818	. . . 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 6052	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐵)) → (𝑓 “ ((iEdg‘𝐴)‘𝑖)) = (( I ↾ (Vtx‘𝐵)) “ ((iEdg‘𝐴)‘𝑖)))
50	49	eqeq1d 2734	. . . . . . 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 3177	. . . . . 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 3588	. 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 2732	. . . 4 ⊢ (Vtx‘𝐵) = (Vtx‘𝐵)
57		eqid 2732	. . . 4 ⊢ (iEdg‘𝐵) = (iEdg‘𝐵)
58	15, 56, 16, 57	isomgr 46477	. . 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 I cid 5572 dom cdm 5675 ↾ cres 5677 “ cima 5678 –1-1-onto→wf1o 6539 ‘cfv 6540 Vtxcvtx 28245 iEdgciedg 28246 UHGraphcuhgr 28305 IsomGr cisomgr 46473

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-uhgr 28307 df-isomgr 46475

This theorem is referenced by: isomgrref 46489 strisomgrop 46494

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