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Mirrors > Home > MPE Home > Th. List > uspgrf1oedg | Structured version Visualization version GIF version |
Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
usgrf1o.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uspgrf1oedg | ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1o.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uspgrf 29186 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
4 | f1f1orn 6860 | . . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸) | |
5 | 2 | rneqi 5951 | . . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺) |
6 | edgval 29081 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
7 | 5, 6 | eqtr4i 2766 | . . . 4 ⊢ ran 𝐸 = (Edg‘𝐺) |
8 | f1oeq3 6839 | . . . 4 ⊢ (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) |
10 | 4, 9 | sylib 218 | . 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) |
11 | 3, 10 | syl 17 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ∅c0 4339 𝒫 cpw 4605 {csn 4631 class class class wbr 5148 dom cdm 5689 ran crn 5690 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 ≤ cle 11294 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 USPGraphcuspgr 29180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-edg 29080 df-uspgr 29182 |
This theorem is referenced by: uspgredgiedg 29207 uspgriedgedg 29208 uspgr2wlkeq 29679 wlkiswwlks2lem4 29902 wlkiswwlks2lem5 29903 clwlkclwwlk 30031 isuspgrim0lem 47809 uspgrlimlem1 47891 uspgrlimlem2 47892 uspgrlimlem3 47893 uspgrlimlem4 47894 uspgrlim 47895 |
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