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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 2733 | . . 3 β’ (VtxβπΊ) = (VtxβπΊ) | |
2 | usgrf1o.e | . . 3 β’ πΈ = (iEdgβπΊ) | |
3 | 1, 2 | uspgrf 28147 | . 2 β’ (πΊ β USPGraph β πΈ:dom πΈβ1-1β{π₯ β (π« (VtxβπΊ) β {β }) β£ (β―βπ₯) β€ 2}) |
4 | f1f1orn 6796 | . . 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 5893 | . . . . 5 β’ ran πΈ = ran (iEdgβπΊ) |
6 | edgval 28042 | . . . . 5 β’ (EdgβπΊ) = ran (iEdgβπΊ) | |
7 | 5, 6 | eqtr4i 2764 | . . . 4 β’ ran πΈ = (EdgβπΊ) |
8 | f1oeq3 6775 | . . . 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 1542 β wcel 2107 {crab 3406 β cdif 3908 β c0 4283 π« cpw 4561 {csn 4587 class class class wbr 5106 dom cdm 5634 ran crn 5635 β1-1βwf1 6494 β1-1-ontoβwf1o 6496 βcfv 6497 β€ cle 11195 2c2 12213 β―chash 14236 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 USPGraphcuspgr 28141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-edg 28041 df-uspgr 28143 |
This theorem is referenced by: uspgr2wlkeq 28636 wlkiswwlks2lem4 28859 wlkiswwlks2lem5 28860 clwlkclwwlk 28988 |
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