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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 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 2748 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1oedg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | usgrf 26220 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
4 | f1f1orn 6297 | . . 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 26111 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
9 | 2 | eqcomi 2757 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼 |
10 | 9 | rneqi 5495 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
11 | 8, 10 | syl6eq 2798 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼) |
12 | 6, 11 | syl5eq 2794 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼) |
13 | 12 | f1oeq3d 6283 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼)) |
14 | 5, 13 | mpbird 247 | 1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1620 ∈ wcel 2127 {crab 3042 ∖ cdif 3700 ∅c0 4046 𝒫 cpw 4290 {csn 4309 dom cdm 5254 ran crn 5255 –1-1→wf1 6034 –1-1-onto→wf1o 6036 ‘cfv 6037 2c2 11233 ♯chash 13282 Vtxcvtx 26044 iEdgciedg 26045 Edgcedg 26109 USGraphcusgr 26214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-edg 26110 df-usgr 26216 |
This theorem is referenced by: usgr2trlncl 26837 |
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