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| Mirrors > Home > MPE Home > Th. List > uspgrf1oedg | Structured version Visualization version GIF version | ||
| Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.) | 
| Ref | Expression | 
|---|---|
| usgrf1o.e | ⊢ 𝐸 = (iEdg‘𝐺) | 
| Ref | Expression | 
|---|---|
| uspgrf1oedg | ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | usgrf1o.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 3 | 1, 2 | uspgrf 29172 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 4 | f1f1orn 6858 | . . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸) | |
| 5 | 2 | rneqi 5947 | . . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺) | 
| 6 | edgval 29067 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 7 | 5, 6 | eqtr4i 2767 | . . . 4 ⊢ ran 𝐸 = (Edg‘𝐺) | 
| 8 | f1oeq3 6837 | . . . 4 ⊢ (ran 𝐸 = (Edg‘𝐺) → (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺))) | |
| 9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) | 
| 10 | 4, 9 | sylib 218 | . 2 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) | 
| 11 | 3, 10 | syl 17 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1-onto→(Edg‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 {csn 4625 class class class wbr 5142 dom cdm 5684 ran crn 5685 –1-1→wf1 6557 –1-1-onto→wf1o 6559 ‘cfv 6560 ≤ cle 11297 2c2 12322 ♯chash 14370 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 USPGraphcuspgr 29166 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-edg 29066 df-uspgr 29168 | 
| This theorem is referenced by: uspgredgiedg 29193 uspgriedgedg 29194 uspgr2wlkeq 29665 wlkiswwlks2lem4 29893 wlkiswwlks2lem5 29894 clwlkclwwlk 30022 isuspgrim0lem 47876 uspgrlimlem1 47960 uspgrlimlem2 47961 uspgrlimlem3 47962 uspgrlimlem4 47963 uspgrlim 47964 | 
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