Theorem uhgrimedg 47895

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 47894	. . 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 2730	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2730	. . . . . . . . 9 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
10	8, 9	grimf1o 47888	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
11		f1of1 6802	. . . . . . . 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 6819	. . . 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 47892	. . . . . . 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 47894	. . . 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 2830	. 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 2109   ⊆ wss 3917  ^◡ccnv 5640   “ cima 5644  –1-1→wf1 6511  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981  UHGraphcuhgr 28990   GraphIso cgrim 47879

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-edg 28982  df-uhgr 28992  df-grim 47882

This theorem is referenced by:  uhgrimprop  47896