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Mirrors > Home > MPE Home > Th. List > umgraf | Structured version Visualization version GIF version |
Description: The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
umgraf | ⊢ ((𝑉 UMGrph 𝐸 ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgraf2 25846 | . . 3 ⊢ (𝑉 UMGrph 𝐸 → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) | |
2 | fndm 5904 | . . . 4 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴) | |
3 | 2 | feq2d 5944 | . . 3 ⊢ (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) |
4 | 1, 3 | syl5ibcom 234 | . 2 ⊢ (𝑉 UMGrph 𝐸 → (𝐸 Fn 𝐴 → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) |
5 | 4 | imp 444 | 1 ⊢ ((𝑉 UMGrph 𝐸 ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 {crab 2900 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 UMGrph cumg 25841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-umgra 25842 |
This theorem is referenced by: umgrass 25848 umgran0 25849 umgrale 25850 umgraun 25857 |
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