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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 2622 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | usgrf1o.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uspgrf 26049 | . 2 ⊢ (𝐺 ∈ USPGraph → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}) |
4 | f1f1orn 6148 | . . 3 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸) | |
5 | 2 | rneqi 5352 | . . . . 5 ⊢ ran 𝐸 = ran (iEdg‘𝐺) |
6 | edgval 25941 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
7 | 5, 6 | eqtr4i 2647 | . . . 4 ⊢ ran 𝐸 = (Edg‘𝐺) |
8 | f1oeq3 6129 | . . . 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 208 | . 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 196 = wceq 1483 ∈ wcel 1990 {crab 2916 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {csn 4177 class class class wbr 4653 dom cdm 5114 ran crn 5115 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 ≤ cle 10075 2c2 11070 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 USPGraph cuspgr 26043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-edg 25940 df-uspgr 26045 |
This theorem is referenced by: uspgr2wlkeq 26542 wlkiswwlks2lem4 26758 wlkiswwlks2lem5 26759 clwlkclwwlk 26903 |
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