Theorem grimidvtxedg 48691

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 6866	. . 3 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
2		grimidvtxsdg.v	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))
3	2	f1oeq3d 6824	. . 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 6575	. . . . 5 ⊢ Fun I
6		fvex 6901	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
7	6	dmex 7915	. . . . 5 ⊢ dom (iEdg‘𝐺) ∈ V
8		resfunexg 7220	. . . . 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 6866	. . . . 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 5900	. . . . . 6 ⊢ (𝜑 → dom (iEdg‘𝐺) = dom (iEdg‘𝐻))
14	13	f1oeq3d 6824	. . . . 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 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = ((iEdg‘𝐺)‘𝑖))
19	12	eqcomd 2772	. . . . . . . 8 ⊢ (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))
20	19	fveq1d 6890	. . . . . . 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 2766	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2766	. . . . . . . . 9 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	23, 24	uhgrss 29451	. . . . . . . 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 6083	. . . . . . 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 2811	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(( I ↾ dom (iEdg‘𝐺))‘𝑖)) = (( I ↾ (Vtx‘𝐺)) “ ((iEdg‘𝐺)‘𝑖)))
30	29	ralrimiva 3160	. . . 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 6815	. . . 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 6887	. . . . . 6 ⊢ (𝑗 = ( I ↾ dom (iEdg‘𝐺)) → (𝑗‘𝑖) = (( I ↾ dom (iEdg‘𝐺))‘𝑖))
34	33	fveqeq2d 6896	. . . . 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 3191	. . . 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 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 6901	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
40		resfunexg 7220	. . . . 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 2766	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
44		eqid 2766	. . . 4 ⊢ (iEdg‘𝐻) = (iEdg‘𝐻)
45	23, 43, 24, 44	isgrim 48688	. . 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 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   ⊆ wss 3908   I cid 5560  dom cdm 5666   ↾ cres 5668   “ cima 5669  Fun wfun 6537  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383  iEdgciedg 29384  UHGraphcuhgr 29443   GraphIso cgrim 48681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835  df-uhgr 29445  df-grim 48684

This theorem is used by:  grimid  48692  opstrgric  48732