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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 29103 | . . 3 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
| 4 | fndm 6671 | . . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴) | |
| 5 | 4 | feq2d 6722 | . . 3 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
| 6 | 3, 5 | syl5ibcom 245 | . 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
| 7 | 6 | imp 406 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 dom cdm 5685 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ≤ cle 11296 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 UPGraphcupgr 29097 |
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-upgr 29099 |
| This theorem is referenced by: upgrn0 29106 upgrle 29107 |
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