Theorem uhgrimedg 48396

Description: An isomorphism between graphs preserves edges, i.e. there is an edge in one graph connecting vertices iff there is an edge in the other graph connecting the corresponding vertices. (Contributed by AV, 25-Oct-2025.)

Hypotheses

Ref	Expression
uhgrimedgi.e	⊢ 𝐸 = (Edg‘𝐺)
uhgrimedgi.d	⊢ 𝐷 = (Edg‘𝐻)

Assertion

Ref	Expression
uhgrimedg	⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))

Proof of Theorem uhgrimedg

Step	Hyp	Ref	Expression
1		simp1 1143	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
2		simp2 1144	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
3	2	anim1i 622	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ 𝐾 ∈ 𝐸) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸))
4		uhgrimedgi.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
5		uhgrimedgi.d	. . . 4 ⊢ 𝐷 = (Edg‘𝐻)
6	4, 5	uhgrimedgi 48395	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)
7	1, 3, 6	syl2an2r 692	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷)
8		eqid 2741	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2741	. . . . . . . . 9 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
10	8, 9	grimf1o 48389	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
11		f1of1 6770	. . . . . . . 8 ⊢ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
13	12	3ad2ant2 1141	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
14		simp3 1145	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐾 ⊆ (Vtx‘𝐺))
15	13, 14	jca 517	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)))
16	15	adantr 482	. . . 4 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)))
17		f1imacnv 6787	. . . 4 ⊢ ((𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (◡𝐹 “ (𝐹 “ 𝐾)) = 𝐾)
18	16, 17	syl 17	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 “ (𝐹 “ 𝐾)) = 𝐾)
19		pm3.22 461	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
20	19	3ad2ant1 1140	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
21		simpl 484	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ∈ UHGraph)
22	21	anim1i 622	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
23	22	3adant3 1139	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
24		grimcnv 48393	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝐹 ∈ (𝐺 GraphIso 𝐻) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺)))
25	24	imp 408	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺))
26	23, 25	syl 17	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺))
27	26	anim1i 622	. . . 4 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 ∈ (𝐻 GraphIso 𝐺) ∧ (𝐹 “ 𝐾) ∈ 𝐷))
28	5, 4	uhgrimedgi 48395	. . . 4 ⊢ (((𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) ∧ (◡𝐹 ∈ (𝐻 GraphIso 𝐺) ∧ (𝐹 “ 𝐾) ∈ 𝐷)) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
29	20, 27, 28	syl2an2r 692	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
30	18, 29	eqeltrrd 2842	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → 𝐾 ∈ 𝐸)
31	7, 30	impbida 807	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885  ^◡ccnv 5620   “ cima 5624  –1-1→wf1 6486  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  Vtxcvtx 29087  Edgcedg 29138  UHGraphcuhgr 29147   GraphIso cgrim 48380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-edg 29139  df-uhgr 29149  df-grim 48383

This theorem is referenced by:  uhgrimprop  48397