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Mirrors > Home > MPE Home > Th. List > upgrfn | Structured version Visualization version GIF version |
Description: The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
Ref | Expression |
---|---|
isupgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isupgr.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgrfn | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isupgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isupgr.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | upgrf 26879 | . . 3 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
4 | fndm 6425 | . . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴) | |
5 | 4 | feq2d 6473 | . . 3 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
6 | 3, 5 | syl5ibcom 248 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
7 | 6 | imp 410 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ∖ cdif 3878 ∅c0 4243 𝒫 cpw 4497 {csn 4525 class class class wbr 5030 dom cdm 5519 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 ≤ cle 10665 2c2 11680 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 UPGraphcupgr 26873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-upgr 26875 |
This theorem is referenced by: upgrn0 26882 upgrle 26883 |
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