![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2725 | . . 3 β’ (VtxβπΊ) = (VtxβπΊ) | |
2 | usgrf1o.e | . . 3 β’ πΈ = (iEdgβπΊ) | |
3 | 1, 2 | uspgrf 29006 | . 2 β’ (πΊ β USPGraph β πΈ:dom πΈβ1-1β{π₯ β (π« (VtxβπΊ) β {β }) β£ (β―βπ₯) β€ 2}) |
4 | f1f1orn 6843 | . . 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 5934 | . . . . 5 β’ ran πΈ = ran (iEdgβπΊ) |
6 | edgval 28901 | . . . . 5 β’ (EdgβπΊ) = ran (iEdgβπΊ) | |
7 | 5, 6 | eqtr4i 2756 | . . . 4 β’ ran πΈ = (EdgβπΊ) |
8 | f1oeq3 6822 | . . . 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 1533 β wcel 2098 {crab 3419 β cdif 3938 β c0 4319 π« cpw 4599 {csn 4625 class class class wbr 5144 dom cdm 5673 ran crn 5674 β1-1βwf1 6540 β1-1-ontoβwf1o 6542 βcfv 6543 β€ cle 11274 2c2 12292 β―chash 14316 Vtxcvtx 28848 iEdgciedg 28849 Edgcedg 28899 USPGraphcuspgr 29000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-edg 28900 df-uspgr 29002 |
This theorem is referenced by: uspgredgiedg 29027 uspgriedgedg 29028 uspgr2wlkeq 29499 wlkiswwlks2lem4 29722 wlkiswwlks2lem5 29723 clwlkclwwlk 29851 isuspgrim0lem 47277 |
Copyright terms: Public domain | W3C validator |