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Mirrors > Home > MPE Home > Th. List > usgrf1oedg | Structured version Visualization version GIF version |
Description: The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by AV, 18-Oct-2020.) |
Ref | Expression |
---|---|
usgrf1oedg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
usgrf1oedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgrf1oedg | ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1oedg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | usgrf 26867 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
4 | f1f1orn 6619 | . . 3 ⊢ (𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) |
6 | usgrf1oedg.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
7 | edgval 26761 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
9 | 2 | eqcomi 2827 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼 |
10 | 9 | rneqi 5800 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
11 | 8, 10 | syl6eq 2869 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼) |
12 | 6, 11 | syl5eq 2865 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼) |
13 | 12 | f1oeq3d 6605 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼)) |
14 | 5, 13 | mpbird 258 | 1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 ∖ cdif 3930 ∅c0 4288 𝒫 cpw 4535 {csn 4557 dom cdm 5548 ran crn 5549 –1-1→wf1 6345 –1-1-onto→wf1o 6347 ‘cfv 6348 2c2 11680 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 Edgcedg 26759 USGraphcusgr 26861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-edg 26760 df-usgr 26863 |
This theorem is referenced by: usgr2trlncl 27468 |
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