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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlsegvdeglem6 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 41395. (Contributed by AV, 21-Feb-2021.) |
Ref | Expression |
---|---|
trlsegvdeg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlsegvdeg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlsegvdeg.f | ⊢ (𝜑 → Fun 𝐼) |
trlsegvdeg.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) |
trlsegvdeg.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
trlsegvdeg.w | ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) |
trlsegvdeg.vx | ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) |
trlsegvdeg.vy | ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) |
trlsegvdeg.vz | ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) |
trlsegvdeg.ix | ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
trlsegvdeg.iy | ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
trlsegvdeg.iz | ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) |
Ref | Expression |
---|---|
trlsegvdeglem6 | ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | trlsegvdeg.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | trlsegvdeg.f | . . 3 ⊢ (𝜑 → Fun 𝐼) | |
4 | trlsegvdeg.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) | |
5 | trlsegvdeg.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
6 | trlsegvdeg.w | . . 3 ⊢ (𝜑 → 𝐹(TrailS‘𝐺)𝑃) | |
7 | trlsegvdeg.vx | . . 3 ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉) | |
8 | trlsegvdeg.vy | . . 3 ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉) | |
9 | trlsegvdeg.vz | . . 3 ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉) | |
10 | trlsegvdeg.ix | . . 3 ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) | |
11 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
12 | trlsegvdeg.iz | . . 3 ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁)))) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | trlsegvdeglem4 41391 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)) |
14 | 2 | trlf1 40906 | . . . . 5 ⊢ (𝐹(TrailS‘𝐺)𝑃 → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼) |
15 | f1fun 6016 | . . . . 5 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼 → Fun 𝐹) | |
16 | 6, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → Fun 𝐹) |
17 | fzofi 12635 | . . . 4 ⊢ (0..^𝑁) ∈ Fin | |
18 | imafi 8142 | . . . 4 ⊢ ((Fun 𝐹 ∧ (0..^𝑁) ∈ Fin) → (𝐹 “ (0..^𝑁)) ∈ Fin) | |
19 | 16, 17, 18 | sylancl 693 | . . 3 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ∈ Fin) |
20 | infi 8069 | . . 3 ⊢ ((𝐹 “ (0..^𝑁)) ∈ Fin → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝜑 → ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼) ∈ Fin) |
22 | 13, 21 | eqeltrd 2688 | 1 ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 {csn 4125 〈cop 4131 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 ...cfz 12197 ..^cfzo 12334 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 TrailSctrls 40899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 df-trls 40901 |
This theorem is referenced by: trlsegvdeg 41395 eupth2lem3lem1 41396 |
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