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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem5 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 29174. (Contributed by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | β’ π = (VtxβπΊ) |
trlsegvdeg.i | β’ πΌ = (iEdgβπΊ) |
trlsegvdeg.f | β’ (π β Fun πΌ) |
trlsegvdeg.n | β’ (π β π β (0..^(β―βπΉ))) |
trlsegvdeg.u | β’ (π β π β π) |
trlsegvdeg.w | β’ (π β πΉ(TrailsβπΊ)π) |
trlsegvdeg.vx | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.vy | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.vz | β’ (π β (Vtxβπ) = π) |
trlsegvdeg.ix | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) |
trlsegvdeg.iy | β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) |
trlsegvdeg.iz | β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) |
Ref | Expression |
---|---|
trlsegvdeglem5 | β’ (π β dom (iEdgβπ) = {(πΉβπ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.iy | . . 3 β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) | |
2 | 1 | dmeqd 5862 | . 2 β’ (π β dom (iEdgβπ) = dom {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) |
3 | fvex 6856 | . . 3 β’ (πΌβ(πΉβπ)) β V | |
4 | dmsnopg 6166 | . . 3 β’ ((πΌβ(πΉβπ)) β V β dom {β¨(πΉβπ), (πΌβ(πΉβπ))β©} = {(πΉβπ)}) | |
5 | 3, 4 | mp1i 13 | . 2 β’ (π β dom {β¨(πΉβπ), (πΌβ(πΉβπ))β©} = {(πΉβπ)}) |
6 | 2, 5 | eqtrd 2777 | 1 β’ (π β dom (iEdgβπ) = {(πΉβπ)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3446 {csn 4587 β¨cop 4593 class class class wbr 5106 dom cdm 5634 βΎ cres 5636 β cima 5637 Fun wfun 6491 βcfv 6497 (class class class)co 7358 0cc0 11052 ...cfz 13425 ..^cfzo 13568 β―chash 14231 Vtxcvtx 27950 iEdgciedg 27951 Trailsctrls 28641 |
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-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-dm 5644 df-iota 6449 df-fv 6505 |
This theorem is referenced by: trlsegvdeglem7 29173 trlsegvdeg 29174 |
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