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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 2735 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1oedg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | usgrf 29187 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
4 | f1f1orn 6860 | . . 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 29081 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
9 | 2 | eqcomi 2744 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼 |
10 | 9 | rneqi 5951 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
11 | 8, 10 | eqtrdi 2791 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼) |
12 | 6, 11 | eqtrid 2787 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼) |
13 | 12 | f1oeq3d 6846 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐼:dom 𝐼–1-1-onto→𝐸 ↔ 𝐼:dom 𝐼–1-1-onto→ran 𝐼)) |
14 | 5, 13 | mpbird 257 | 1 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1-onto→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 ∅c0 4339 𝒫 cpw 4605 {csn 4631 dom cdm 5689 ran crn 5690 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 2c2 12319 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 USGraphcusgr 29181 |
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-usgr 29183 |
This theorem is referenced by: usgr2trlncl 29793 |
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