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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 26938 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USGraph ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})) |
4 | 3 | ibi 269 | 1 ⊢ (𝐺 ∈ USGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 ∖ cdif 3933 ∅c0 4291 𝒫 cpw 4539 {csn 4567 dom cdm 5555 –1-1→wf1 6352 ‘cfv 6355 2c2 11693 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 USGraphcusgr 26934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fv 6363 df-usgr 26936 |
This theorem is referenced by: usgredg2ALT 26975 usgrf1oedg 26989 usgrsizedg 26997 usgrres 27090 |
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