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Mirrors > Home > MPE Home > Th. List > isusgra | Structured version Visualization version GIF version |
Description: The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.) |
Ref | Expression |
---|---|
isusgra | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1eq1 6009 | . . . 4 ⊢ (𝑒 = 𝐸 → (𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2})) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
3 | dmeq 5246 | . . . . 5 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸) | |
4 | 3 | adantl 481 | . . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸) |
5 | f1eq2 6010 | . . . 4 ⊢ (dom 𝑒 = dom 𝐸 → (𝐸:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2})) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝐸:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
7 | simpl 472 | . . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉) | |
8 | 7 | pweqd 4113 | . . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉) |
9 | 8 | difeq1d 3689 | . . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
10 | rabeq 3166 | . . . 4 ⊢ ((𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) | |
11 | f1eq3 6011 | . . . 4 ⊢ ({𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) | |
12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
13 | 2, 6, 12 | 3bitrd 293 | . 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
14 | df-usgra 25862 | . 2 ⊢ USGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}} | |
15 | 13, 14 | brabga 4914 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉 USGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ 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 –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-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-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-usgra 25862 |
This theorem is referenced by: usgraf 25875 isusgra0 25876 usgraeq12d 25891 usisuslgra 25894 usgrares 25898 usgra0 25899 usgra0v 25900 usgra1v 25919 |
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