Theorem grimidvtxedg 48471

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 6841	. . 3 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
2		grimidvtxsdg.v	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
3	2	f1oeq3d 6799	. . 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 235	. 2 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		funi 6549	. . . . 5 ⊢ Fun I
6		fvex 6876	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7	6	dmex 7886	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
8		resfunexg 7195	. . . . 5 ⊢ ((Fun I ∧ dom (iEdg‘𝐺) ∈ V) → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
9	5, 7, 8	mp2an 702	. . . 4 ⊢ ( I ↾ dom (iEdg‘𝐺)) ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
11		f1oi 6841	. . . . 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 5879	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom (iEdg‘𝐻))
14	13	f1oeq3d 6799	. . . . 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 235	. . . 4 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
16		fvresi 7153	. . . . . . . 8 ⊢ (𝑖 ∈ dom (iEdg‘𝐺) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
17	16	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
18	17	fveq2d 6867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
19	12	eqcomd 2767	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
20	19	fveq1d 6865	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)))
21	20	adantr 484	. . . . . 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 2761	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2761	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	23, 24	uhgrss 29211	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
26	22, 25	sylan 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
27		resiima 6062	. . . . . . 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 2806	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
30	29	ralrimiva 3153	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
31	15, 30	jca 519	. . 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 6790	. . . 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 6862	. . . . . 6 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗‘𝑖) = (( I ↾ dom (iEdg‘𝐺))‘𝑖))
34	33	fveqeq2d 6871	. . . . 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 3184	. . . 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 641	. . 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 3557	. 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 6876	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
40		resfunexg 7195	. . . . 5 ⊢ ((Fun I ∧ (Vtx‘𝐺) ∈ V) → ( I ↾ (Vtx‘𝐺)) ∈ V)
41	5, 39, 40	mp2an 702	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)) ∈ V
42	41	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ V)
43		eqid 2761	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
44		eqid 2761	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
45	23, 43, 24, 44	isgrim 48468	. . 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 1389	. 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 723	1 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   I cid 5539  dom cdm 5645   ↾ cres 5647   “ cima 5648  Fun wfun 6511  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143  iEdgciedg 29144  UHGraphcuhgr 29203   GraphIso cgrim 48461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-map 8805  df-uhgr 29205  df-grim 48464

This theorem is referenced by:  grimid  48472  opstrgric  48512