Theorem grimidvtxedg 48505

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 6849	. . 3 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
2		grimidvtxsdg.v	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
3	2	f1oeq3d 6807	. . 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 236	. 2 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		funi 6557	. . . . 5 ⊢ Fun I
6		fvex 6884	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7	6	dmex 7894	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
8		resfunexg 7203	. . . . 5 ⊢ ((Fun I ∧ dom (iEdg‘𝐺) ∈ V) → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
9	5, 7, 8	mp2an 704	. . . 4 ⊢ ( I ↾ dom (iEdg‘𝐺)) ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
11		f1oi 6849	. . . . 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 5886	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom (iEdg‘𝐻))
14	13	f1oeq3d 6807	. . . . 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 236	. . . 4 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
16		fvresi 7161	. . . . . . . 8 ⊢ (𝑖 ∈ dom (iEdg‘𝐺) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
17	16	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
18	17	fveq2d 6875	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
19	12	eqcomd 2771	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
20	19	fveq1d 6873	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)))
21	20	adantr 485	. . . . . 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 2765	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2765	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	23, 24	uhgrss 29323	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
26	22, 25	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
27		resiima 6069	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
28	26, 27	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
29	18, 21, 28	3eqtr4d 2810	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
30	29	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
31	15, 30	jca 520	. . 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 6798	. . . 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 6870	. . . . . 6 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗‘𝑖) = (( I ↾ dom (iEdg‘𝐺))‘𝑖))
34	33	fveqeq2d 6879	. . . . 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 3188	. . . 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 643	. . 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 3560	. 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 6884	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
40		resfunexg 7203	. . . . 5 ⊢ ((Fun I ∧ (Vtx‘𝐺) ∈ V) → ( I ↾ (Vtx‘𝐺)) ∈ V)
41	5, 39, 40	mp2an 704	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)) ∈ V
42	41	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ V)
43		eqid 2765	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
44		eqid 2765	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
45	23, 43, 24, 44	isgrim 48502	. . 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 1394	. 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 725	1 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907   I cid 5546  dom cdm 5652   ↾ cres 5654   “ cima 5655  Fun wfun 6519  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Vtxcvtx 29255  iEdgciedg 29256  UHGraphcuhgr 29315   GraphIso cgrim 48495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814  df-uhgr 29317  df-grim 48498

This theorem is referenced by:  grimid  48506  opstrgric  48546