Theorem grimidvtxedg 49076

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 6863	. . 3 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
2		grimidvtxsdg.v	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
3	2	f1oeq3d 6821	. . 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 6572	. . . . 5 ⊢ Fun I
6		fvex 6898	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7	6	dmex 7921	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
8		resfunexg 7221	. . . . 5 ⊢ ((Fun I ∧ dom (iEdg‘𝐺) ∈ V) → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
9	5, 7, 8	mp2an 705	. . . 4 ⊢ ( I ↾ dom (iEdg‘𝐺)) ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ dom (iEdg‘𝐺)) ∈ V)
11		f1oi 6863	. . . . 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 6821	. . . . 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 7178	. . . . . . . 8 ⊢ (𝑖 ∈ dom (iEdg‘𝐺) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
17	16	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → (( I ↾ dom (iEdg‘𝐺))‘𝑖) = 𝑖)
18	17	fveq2d 6889	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
19	12	eqcomd 2766	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
20	19	fveq1d 6887	. . . . . . 7 ⊢ (𝜑 → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)))
21	20	adantr 486	. . . . . 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 2760	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2760	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	23, 24	uhgrss 29647	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
26	22, 25	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑖) ⊆ (Vtx‘𝐺))
27		resiima 6073	. . . . . . 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 2805	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
30	29	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
31	15, 30	jca 521	. . 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 6812	. . . 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 6884	. . . . . 6 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗‘𝑖) = (( I ↾ dom (iEdg‘𝐺))‘𝑖))
34	33	fveqeq2d 6893	. . . . 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 3185	. . . 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 644	. . 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 3552	. 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 6898	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
40		resfunexg 7221	. . . . 5 ⊢ ((Fun I ∧ (Vtx‘𝐺) ∈ V) → ( I ↾ (Vtx‘𝐺)) ∈ V)
41	5, 39, 40	mp2an 705	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)) ∈ V
42	41	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ V)
43		eqid 2760	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
44		eqid 2760	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
45	23, 43, 24, 44	isgrim 49073	. . 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 1398	. 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 726	1 ⊢ (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   I cid 5544  dom cdm 5650   ↾ cres 5652   “ cima 5653  Fun wfun 6532  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29579  iEdgciedg 29580  UHGraphcuhgr 29639   GraphIso cgrim 49066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8852  df-uhgr 29641  df-grim 49069

This theorem is used by:  grimid  49077  opstrgric  49117