Theorem isuspgrim 47851

Description: A class is an isomorphism of simple pseudographs iff it is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. This corresponds to the formal definition in [Bollobas] p. 3 and the definition in [Diestel] p. 3. (Contributed by AV, 27-Apr-2025.)

Hypotheses

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

Assertion

Ref	Expression
isuspgrim	⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (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,𝐸   𝑥,𝐹   𝑥,𝐷,𝑦   𝑦,𝐸   𝑦,𝐹   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑉,𝑦

Allowed substitution hints:   𝑊(𝑥,𝑦)

Proof of Theorem isuspgrim

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgruhgr 29118	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
2		uspgruhgr 29118	. . . . . . 7 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UHGraph)
3	1, 2	anim12i 613	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph))
4	3	anim1i 615	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
5		df-3an 1088	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ↔ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
6	4, 5	sylibr 234	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
7		isusgrim.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
8		isusgrim.d	. . . . 5 ⊢ 𝐷 = (Edg‘𝐻)
9		isusgrim.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10		isusgrim.w	. . . . 5 ⊢ 𝑊 = (Vtx‘𝐻)
11	7, 8, 9, 10	uhgrimprop 47847	. . . 4 ⊢ ((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)))
12	6, 11	syl 17	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)))
13	12	ex 412	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
14		f1of 6807	. . . . . 6 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉⟶𝑊)
15	9	fvexi 6879	. . . . . . 7 ⊢ 𝑉 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝑉 ∈ V)
17	14, 16	fexd 7208	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹 ∈ V)
18	17	adantl 481	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → 𝐹 ∈ V)
19		simpllr 775	. . . . . 6 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝐹 ∈ V) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → 𝐹:𝑉–1-1-onto→𝑊)
20	9, 10, 7, 8	isuspgrimlem 47850	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)
21	20	adantlr 715	. . . . . 6 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝐹 ∈ V) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)
22	9, 10, 7, 8	isuspgrim0 47849	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ V) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
23	22	ad5ant124 1367	. . . . . 6 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝐹 ∈ V) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
24	19, 21, 23	mpbir2and 713	. . . . 5 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝐹 ∈ V) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
25	24	ex 412	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝐹 ∈ V) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → 𝐹 ∈ (𝐺 GraphIso 𝐻)))
26	18, 25	mpdan 687	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷) → 𝐹 ∈ (𝐺 GraphIso 𝐻)))
27	26	expimpd 453	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → ((𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → 𝐹 ∈ (𝐺 GraphIso 𝐻)))
28	13, 27	impbid 212	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3046  Vcvv 3455  {cpr 4599   ↦ cmpt 5196   “ cima 5649  –1-1-onto→wf1o 6518  ‘cfv 6519  (class class class)co 7394  Vtxcvtx 28930  Edgcedg 28981  UHGraphcuhgr 28990  USPGraphcuspgr 29082   GraphIso cgrim 47830

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-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718  ax-cnex 11142  ax-resscn 11143  ax-1cn 11144  ax-icn 11145  ax-addcl 11146  ax-addrcl 11147  ax-mulcl 11148  ax-mulrcl 11149  ax-mulcom 11150  ax-addass 11151  ax-mulass 11152  ax-distr 11153  ax-i2m1 11154  ax-1ne0 11155  ax-1rid 11156  ax-rnegex 11157  ax-rrecex 11158  ax-cnre 11159  ax-pre-lttri 11160  ax-pre-lttrn 11161  ax-pre-ltadd 11162  ax-pre-mulgt0 11163

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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 2880  df-ne 2928  df-nel 3032  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-int 4919  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-1st 7977  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-1o 8443  df-2o 8444  df-oadd 8447  df-er 8682  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-dju 9872  df-card 9910  df-pnf 11228  df-mnf 11229  df-xr 11230  df-ltxr 11231  df-le 11232  df-sub 11425  df-neg 11426  df-nn 12198  df-2 12260  df-n0 12459  df-xnn0 12532  df-z 12546  df-uz 12810  df-fz 13482  df-hash 14306  df-edg 28982  df-uhgr 28992  df-upgr 29016  df-uspgr 29084  df-grim 47833

This theorem is referenced by:  gricuspgr  47873