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Mirrors > Home > MPE Home > Th. List > usgraedg2 | Structured version Visualization version GIF version |
Description: The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 25850. (Contributed by Alexander van der Vekens, 11-Aug-2017.) |
Ref | Expression |
---|---|
usgraedg2 | ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ dom 𝐸) → (#‘(𝐸‘𝑋)) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgraf 25875 | . . . 4 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) | |
2 | f1f 6014 | . . . 4 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) |
4 | 3 | ffvelrnda 6267 | . 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ dom 𝐸) → (𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) |
5 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = (𝐸‘𝑋) → (#‘𝑥) = (#‘(𝐸‘𝑋))) | |
6 | 5 | eqeq1d 2612 | . . . 4 ⊢ (𝑥 = (𝐸‘𝑋) → ((#‘𝑥) = 2 ↔ (#‘(𝐸‘𝑋)) = 2)) |
7 | 6 | elrab 3331 | . . 3 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ ((𝐸‘𝑋) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘(𝐸‘𝑋)) = 2)) |
8 | 7 | simprbi 479 | . 2 ⊢ ((𝐸‘𝑋) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → (#‘(𝐸‘𝑋)) = 2) |
9 | 4, 8 | syl 17 | 1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ dom 𝐸) → (#‘(𝐸‘𝑋)) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 2c2 10947 #chash 12979 USGrph cusg 25859 |
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-sbc 3403 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-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 df-usgra 25862 |
This theorem is referenced by: usgraedgprv 25905 usgranloopv 25907 |
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