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Mirrors > Home > MPE Home > Th. List > uspgrf | Structured version Visualization version GIF version |
Description: The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
isuspgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isuspgr.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uspgrf | ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isuspgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isuspgr.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 27503 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
4 | 3 | ibi 266 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 {crab 3069 ∖ cdif 3888 ∅c0 4261 𝒫 cpw 4538 {csn 4566 class class class wbr 5078 dom cdm 5588 –1-1→wf1 6427 ‘cfv 6430 ≤ cle 10994 2c2 12011 ♯chash 14025 Vtxcvtx 27347 iEdgciedg 27348 USPGraphcuspgr 27499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-nul 5233 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fv 6438 df-uspgr 27501 |
This theorem is referenced by: uspgrf1oedg 27524 usgrumgruspgr 27531 usgruspgrb 27532 usgrislfuspgr 27535 uspgrn2crct 28152 |
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