Theorem grimidvtxedg 47876

Description: The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and a graph with the same vertices and edges. (Contributed by AV, 4-May-2025.)

Hypotheses

Ref	Expression
grimidvtxsdg.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)
grimidvtxsdg.h	⊢ (𝜑 → 𝐻 ∈ 𝑉)
grimidvtxsdg.v	⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
grimidvtxsdg.e	⊢ (𝜑 → (iEdg‘𝐺) = (iEdg‘𝐻))

Assertion

Ref	Expression
grimidvtxedg	⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))

Proof of Theorem grimidvtxedg

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6886	. . 3 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
2		grimidvtxsdg.v	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
3	2	f1oeq3d 6845	. . 3 ⊢ (𝜑 → (( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ↔ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
4	1, 3	mpbii 233	. 2 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		funi 6598	. . . . 5 ⊢ Fun I
6		fvex 6919	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7	6	dmex 7931	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
8		resfunexg 7235	. . . . 5 ⊢ ((Fun I ∧ dom (iEdg‘𝐺) ∈ V) → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
9	5, 7, 8	mp2an 692	. . . 4 ⊢ ( I ↾ dom (iEdg‘𝐺)) ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
11		f1oi 6886	. . . . 5 ⊢ ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐺)
12		grimidvtxsdg.e	. . . . . . 7 ⊢ (𝜑 → (iEdg‘𝐺) = (iEdg‘𝐻))
13	12	dmeqd 5916	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom (iEdg‘𝐻))
14	13	f1oeq3d 6845	. . . . 5 ⊢ (𝜑 → (( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐺) ↔ ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
15	11, 14	mpbii 233	. . . 4 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
16		fvresi 7193	. . . . . . . 8 ⊢ (𝑖 ∈ dom (iEdg‘𝐺) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
18	17	fveq2d 6910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
19	12	eqcomd 2743	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
20	19	fveq1d 6908	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)))
21	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)))
22		grimidvtxsdg.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
23		eqid 2737	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2737	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	23, 24	uhgrss 29081	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
26	22, 25	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
27		resiima 6094	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
29	18, 21, 28	3eqtr4d 2787	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
30	29	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
31	15, 30	jca 511	. . 3 ⊢ (𝜑 → (( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))
32		f1oeq1 6836	. . . 4 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
33		fveq1 6905	. . . . . 6 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗‘𝑖) = (( I ↾ dom (iEdg‘𝐺))‘𝑖))
34	33	fveqeq2d 6914	. . . . 5 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))
35	34	ralbidv 3178	. . . 4 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))
36	32, 35	anbi12d 632	. . 3 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))) ↔ (( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))))
37	10, 31, 36	spcedv 3598	. 2 ⊢ (𝜑 → ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))
38		grimidvtxsdg.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑉)
39		fvex 6919	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
40		resfunexg 7235	. . . . 5 ⊢ ((Fun I ∧ (Vtx‘𝐺) ∈ V) → ( I ↾ (Vtx‘𝐺)) ∈ V)
41	5, 39, 40	mp2an 692	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)) ∈ V
42	41	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ V)
43		eqid 2737	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
44		eqid 2737	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
45	23, 43, 24, 44	isgrim 47868	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ 𝑉 ∧ ( I ↾ (Vtx‘𝐺)) ∈ V) → (( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻) ↔ (( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))))
46	22, 38, 42, 45	syl3anc 1373	. 2 ⊢ (𝜑 → (( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻) ↔ (( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖))))))
47	4, 37, 46	mpbir2and 713	1 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ⊆ wss 3951   I cid 5577  dom cdm 5685   ↾ cres 5687   “ cima 5688  Fun wfun 6555  –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-rep 5279  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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  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-iun 4993  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:  grimid  47877  opstrgric  47895