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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem5 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28009. (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 |
---|---|
trlsegvdeglem5 | ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlsegvdeg.iy | . . 3 ⊢ (𝜑 → (iEdg‘𝑌) = {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) | |
2 | 1 | dmeqd 5777 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
3 | fvex 6686 | . . 3 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
4 | dmsnopg 6073 | . . 3 ⊢ ((𝐼‘(𝐹‘𝑁)) ∈ V → dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {(𝐹‘𝑁)}) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉} = {(𝐹‘𝑁)}) |
6 | 2, 5 | eqtrd 2859 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 class class class wbr 5069 dom cdm 5558 ↾ cres 5560 “ cima 5561 Fun wfun 6352 ‘cfv 6358 (class class class)co 7159 0cc0 10540 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Vtxcvtx 26784 iEdgciedg 26785 Trailsctrls 27475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-dm 5568 df-iota 6317 df-fv 6366 |
This theorem is referenced by: trlsegvdeglem7 28008 trlsegvdeg 28009 |
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