Theorem uhgrimprop 48990

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 48982	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
4	3	3ad2ant3 1153	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹:𝑉–1-1-onto→𝑊)
5		3simpa 1166	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
6		simp3 1156	. . . . 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 3948	. . . . 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 48989	. . . . 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 1449	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ (𝐹 “ {𝑥, 𝑦}) ∈ 𝐷))
13		f1ofn 6825	. . . . . . . . . 10 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 Fn 𝑉)
14	3, 13	syl 18	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹 Fn 𝑉)
15	14	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 Fn 𝑉)
16	15	anim1i 627	. . . . . . 7 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
17		3anass 1111	. . . . . . 7 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)))
18	16, 17	sylibr 237	. . . . . 6 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉))
19		fnimapr 6968	. . . . . 6 ⊢ ((𝐹 Fn 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
20	18, 19	syl 18	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝐹 “ {𝑥, 𝑦}) = {(𝐹‘𝑥), (𝐹‘𝑦)})
21	20	eleq1d 2845	. . . 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 282	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))
23	22	ralrimivva 3205	. 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 521	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  {cpr 4585   “ cima 5653   Fn wfn 6533  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574  Edgcedg 29625  UHGraphcuhgr 29634   GraphIso cgrim 48973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-edg 29626  df-uhgr 29636  df-grim 48976

This theorem is used by:  isuspgrim  48994