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Mirrors > Home > MPE Home > Th. List > usgrf | Structured version Visualization version GIF version |
Description: The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
isuspgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isuspgr.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
usgrf | ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isuspgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isuspgr.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isusgr 26048 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})) |
4 | 3 | ibi 256 | 1 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {csn 4177 dom cdm 5114 –1-1→wf1 5885 ‘cfv 5888 2c2 11070 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 USGraph cusgr 26044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fv 5896 df-usgr 26046 |
This theorem is referenced by: usgredg2ALT 26085 usgrf1oedg 26099 usgrsizedg 26107 usgrres 26200 clwlksfclwwlk 26962 |
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