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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem6 | 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 |
---|---|
trlsegvdeglem6 | β’ (π β dom (iEdgβπ) β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.v | . . 3 β’ π = (VtxβπΊ) | |
2 | trlsegvdeg.i | . . 3 β’ πΌ = (iEdgβπΊ) | |
3 | trlsegvdeg.f | . . 3 β’ (π β Fun πΌ) | |
4 | trlsegvdeg.n | . . 3 β’ (π β π β (0..^(β―βπΉ))) | |
5 | trlsegvdeg.u | . . 3 β’ (π β π β π) | |
6 | trlsegvdeg.w | . . 3 β’ (π β πΉ(TrailsβπΊ)π) | |
7 | trlsegvdeg.vx | . . 3 β’ (π β (Vtxβπ) = π) | |
8 | trlsegvdeg.vy | . . 3 β’ (π β (Vtxβπ) = π) | |
9 | trlsegvdeg.vz | . . 3 β’ (π β (Vtxβπ) = π) | |
10 | trlsegvdeg.ix | . . 3 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0..^π)))) | |
11 | trlsegvdeg.iy | . . 3 β’ (π β (iEdgβπ) = {β¨(πΉβπ), (πΌβ(πΉβπ))β©}) | |
12 | trlsegvdeg.iz | . . 3 β’ (π β (iEdgβπ) = (πΌ βΎ (πΉ β (0...π)))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | trlsegvdeglem4 29170 | . 2 β’ (π β dom (iEdgβπ) = ((πΉ β (0..^π)) β© dom πΌ)) |
14 | 2 | trlf1 28649 | . . . . 5 β’ (πΉ(TrailsβπΊ)π β πΉ:(0..^(β―βπΉ))β1-1βdom πΌ) |
15 | f1fun 6741 | . . . . 5 β’ (πΉ:(0..^(β―βπΉ))β1-1βdom πΌ β Fun πΉ) | |
16 | 6, 14, 15 | 3syl 18 | . . . 4 β’ (π β Fun πΉ) |
17 | fzofi 13880 | . . . 4 β’ (0..^π) β Fin | |
18 | imafi 9120 | . . . 4 β’ ((Fun πΉ β§ (0..^π) β Fin) β (πΉ β (0..^π)) β Fin) | |
19 | 16, 17, 18 | sylancl 587 | . . 3 β’ (π β (πΉ β (0..^π)) β Fin) |
20 | infi 9213 | . . 3 β’ ((πΉ β (0..^π)) β Fin β ((πΉ β (0..^π)) β© dom πΌ) β Fin) | |
21 | 19, 20 | syl 17 | . 2 β’ (π β ((πΉ β (0..^π)) β© dom πΌ) β Fin) |
22 | 13, 21 | eqeltrd 2838 | 1 β’ (π β dom (iEdgβπ) β Fin) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β© cin 3910 {csn 4587 β¨cop 4593 class class class wbr 5106 dom cdm 5634 βΎ cres 5636 β cima 5637 Fun wfun 6491 β1-1βwf1 6494 βcfv 6497 (class class class)co 7358 Fincfn 8884 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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 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-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 df-hash 14232 df-word 14404 df-wlks 28550 df-trls 28643 |
This theorem is referenced by: trlsegvdeg 29174 eupth2lem3lem1 29175 |
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