Theorem uhgrimprop 48519

Description: An isomorphism between hypergraphs is a bijection between their vertices that preserves adjacency for simple edges, i.e. there is a simple edge in one graph connecting one or two vertices iff there is a simple edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.) (Revised by AV, 25-Oct-2025.)

Hypotheses

Ref	Expression
uhgrimedgi.e	⊢ 𝐸 = (Edg‘𝐺)
uhgrimedgi.d	⊢ 𝐷 = (Edg‘𝐻)
uhgrimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrimprop.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
uhgrimprop	⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑦,𝑉

Allowed substitution hints:   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝑉(𝑥)   𝑊(𝑥,𝑦)

Proof of Theorem uhgrimprop

Step	Hyp	Ref	Expression
1		uhgrimprop.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrimprop.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐻)
3	1, 2	grimf1o 48511	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
4	3	3ad2ant3 1149	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹:𝑉–1-1-onto→𝑊)
5		3simpa 1162	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
6		simp3 1152	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
7		prssi 4781	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑥, 𝑦} ⊆ 𝑉)
8	7, 1	sseqtrdi 3978	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑥, 𝑦} ⊆ (Vtx‘𝐺))
9		uhgrimedgi.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
10		uhgrimedgi.d	. . . . . 6 ⊢ 𝐷 = (Edg‘𝐻)
11	9, 10	uhgrimedg 48518	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ {𝑥, 𝑦} ⊆ (Vtx‘𝐺)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷))
12	5, 6, 8, 11	syl2an3an 1443	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷))
13		f1ofn 6809	. . . . . . . . . 10 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
14	3, 13	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹 Fn 𝑉)
15	14	3ad2ant3 1149	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 Fn 𝑉)
16	15	anim1i 624	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
17		3anass 1107	. . . . . . 7 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
18	16, 17	sylibr 236	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
19		fnimapr 6952	. . . . . 6 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
20	18, 19	syl 17	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
21	20	eleq1d 2849	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝐹 “ {𝑥, 𝑦}) ∈ 𝐷 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
22	12, 21	bitrd 281	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
23	22	ralrimivva 3207	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
24	4, 23	jca 519	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  {cpr 4586   “ cima 5652   Fn wfn 6518  –1-1-onto→wf1o 6522  ‘cfv 6523  (class class class)co 7398  Vtxcvtx 29199  Edgcedg 29250  UHGraphcuhgr 29259   GraphIso cgrim 48502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-edg 29251  df-uhgr 29261  df-grim 48505

This theorem is referenced by:  isuspgrim  48523