Theorem uhgrimedg 47852

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 1136	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
2		simp2 1137	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
3	2	anim1i 615	. . 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 47851	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ (𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ∈ 𝐸)) → (𝐹 “ 𝐾) ∈ 𝐷)
7	1, 3, 6	syl2an2r 685	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ 𝐾 ∈ 𝐸) → (𝐹 “ 𝐾) ∈ 𝐷)
8		eqid 2735	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2735	. . . . . . . . 9 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
10	8, 9	grimf1o 47845	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
11		f1of1 6816	. . . . . . . 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 1134	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐹:(Vtx‘𝐺)–1-1→(Vtx‘𝐻))
14		simp3 1138	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → 𝐾 ⊆ (Vtx‘𝐺))
15	13, 14	jca 511	. . . . 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 480	. . . 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 6833	. . . 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		pm3.22 459	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
20	19	3ad2ant1 1133	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph))
21		simpl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → 𝐺 ∈ UHGraph)
22	21	anim1i 615	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
23	22	3adant3 1132	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
24		grimcnv 47849	. . . . . . 7 ⊢ (𝐺 ∈ UHGraph → (𝐹 ∈ (𝐺 GraphIso 𝐻) → ◡𝐹 ∈ (𝐻 GraphIso 𝐺)))
25	24	imp 406	. . . . . 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 615	. . . 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 47851	. . . 4 ⊢ (((𝐻 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) ∧ (◡𝐹 ∈ (𝐻 GraphIso 𝐺) ∧ (𝐹 “ 𝐾) ∈ 𝐷)) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
29	20, 27, 28	syl2an2r 685	. . 3 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → (◡𝐹 “ (𝐹 “ 𝐾)) ∈ 𝐸)
30	18, 29	eqeltrrd 2835	. 2 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) ∧ (𝐹 “ 𝐾) ∈ 𝐷) → 𝐾 ∈ 𝐸)
31	7, 30	impbida 800	1 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝐾 ⊆ (Vtx‘𝐺)) → (𝐾 ∈ 𝐸 ↔ (𝐹 “ 𝐾) ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ^◡ccnv 5653   “ cima 5657  –1-1→wf1 6527  –1-1-onto→wf1o 6529  ‘cfv 6530  (class class class)co 7403  Vtxcvtx 28921  Edgcedg 28972  UHGraphcuhgr 28981   GraphIso cgrim 47836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-map 8840  df-edg 28973  df-uhgr 28983  df-grim 47839

This theorem is referenced by:  uhgrimprop  47853