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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 2731 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1oedg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | usgrf 28848 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
4 | f1f1orn 6844 | . . 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 28742 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
9 | 2 | eqcomi 2740 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼 |
10 | 9 | rneqi 5936 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
11 | 8, 10 | eqtrdi 2787 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼) |
12 | 6, 11 | eqtrid 2783 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼) |
13 | 12 | f1oeq3d 6830 | . 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 1540 ∈ wcel 2105 {crab 3431 ∖ cdif 3945 ∅c0 4322 𝒫 cpw 4602 {csn 4628 dom cdm 5676 ran crn 5677 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 2c2 12274 ♯chash 14297 Vtxcvtx 28689 iEdgciedg 28690 Edgcedg 28740 USGraphcusgr 28842 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-edg 28741 df-usgr 28844 |
This theorem is referenced by: usgr2trlncl 29450 |
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