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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 | β’ πΈ = (iEdgβπΊ) |
Ref | Expression |
---|---|
uspgrf1oedg | β’ (πΊ β USPGraph β πΈ:dom πΈβ1-1-ontoβ(EdgβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . . 3 β’ (VtxβπΊ) = (VtxβπΊ) | |
2 | usgrf1o.e | . . 3 β’ πΈ = (iEdgβπΊ) | |
3 | 1, 2 | uspgrf 28941 | . 2 β’ (πΊ β USPGraph β πΈ:dom πΈβ1-1β{π₯ β (π« (VtxβπΊ) β {β }) β£ (β―βπ₯) β€ 2}) |
4 | f1f1orn 6844 | . . 3 β’ (πΈ:dom πΈβ1-1β{π₯ β (π« (VtxβπΊ) β {β }) β£ (β―βπ₯) β€ 2} β πΈ:dom πΈβ1-1-ontoβran πΈ) | |
5 | 2 | rneqi 5933 | . . . . 5 β’ ran πΈ = ran (iEdgβπΊ) |
6 | edgval 28836 | . . . . 5 β’ (EdgβπΊ) = ran (iEdgβπΊ) | |
7 | 5, 6 | eqtr4i 2758 | . . . 4 β’ ran πΈ = (EdgβπΊ) |
8 | f1oeq3 6823 | . . . 4 β’ (ran πΈ = (EdgβπΊ) β (πΈ:dom πΈβ1-1-ontoβran πΈ β πΈ:dom πΈβ1-1-ontoβ(EdgβπΊ))) | |
9 | 7, 8 | ax-mp 5 | . . 3 β’ (πΈ:dom πΈβ1-1-ontoβran πΈ β πΈ:dom πΈβ1-1-ontoβ(EdgβπΊ)) |
10 | 4, 9 | sylib 217 | . 2 β’ (πΈ:dom πΈβ1-1β{π₯ β (π« (VtxβπΊ) β {β }) β£ (β―βπ₯) β€ 2} β πΈ:dom πΈβ1-1-ontoβ(EdgβπΊ)) |
11 | 3, 10 | syl 17 | 1 β’ (πΊ β USPGraph β πΈ:dom πΈβ1-1-ontoβ(EdgβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1534 β wcel 2099 {crab 3427 β cdif 3941 β c0 4318 π« cpw 4598 {csn 4624 class class class wbr 5142 dom cdm 5672 ran crn 5673 β1-1βwf1 6539 β1-1-ontoβwf1o 6541 βcfv 6542 β€ cle 11265 2c2 12283 β―chash 14307 Vtxcvtx 28783 iEdgciedg 28784 Edgcedg 28834 USPGraphcuspgr 28935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-edg 28835 df-uspgr 28937 |
This theorem is referenced by: uspgredgiedg 28962 uspgriedgedg 28963 uspgr2wlkeq 29434 wlkiswwlks2lem4 29657 wlkiswwlks2lem5 29658 clwlkclwwlk 29786 isuspgrim0lem 47082 |
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