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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem3 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 29460. (Contributed by AV, 20-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 |
---|---|
trlsegvdeglem3 | ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6901 | . . . 4 ⊢ (𝐹‘𝑁) ∈ V | |
2 | fvex 6901 | . . . 4 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
3 | 1, 2 | pm3.2i 472 | . . 3 ⊢ ((𝐹‘𝑁) ∈ V ∧ (𝐼‘(𝐹‘𝑁)) ∈ V) |
4 | funsng 6596 | . . 3 ⊢ (((𝐹‘𝑁) ∈ V ∧ (𝐼‘(𝐹‘𝑁)) ∈ V) → Fun {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → Fun {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
6 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
7 | 6 | funeqd 6567 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑌) ↔ Fun {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
8 | 5, 7 | mpbird 257 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4627 〈cop 4633 class class class wbr 5147 ↾ cres 5677 “ cima 5678 Fun wfun 6534 ‘cfv 6540 (class class class)co 7404 0cc0 11106 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 Vtxcvtx 28236 iEdgciedg 28237 Trailsctrls 28927 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-iota 6492 df-fun 6542 df-fv 6548 |
This theorem is referenced by: trlsegvdeg 29460 |
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