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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 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | usgrf1oedg.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 3 | 1, 2 | usgrf 29172 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
| 4 | f1f1orn 6859 | . . 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 29066 | . . . . . 6 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 8 | 7 | a1i 11 | . . . . 5 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
| 9 | 2 | eqcomi 2746 | . . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼 |
| 10 | 9 | rneqi 5948 | . . . . 5 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
| 11 | 8, 10 | eqtrdi 2793 | . . . 4 ⊢ (𝐺 ∈ USGraph → (Edg‘𝐺) = ran 𝐼) |
| 12 | 6, 11 | eqtrid 2789 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐸 = ran 𝐼) |
| 13 | 12 | f1oeq3d 6845 | . 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 2108 {crab 3436 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 dom cdm 5685 ran crn 5686 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 USGraphcusgr 29166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-edg 29065 df-usgr 29168 |
| This theorem is referenced by: usgr2trlncl 29780 |
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