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Mirrors > Home > MPE Home > Th. List > trlsegvdeglem5 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 28006. (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 5774 | . 2 ⊢ (𝜑 → dom (iEdg‘𝑌) = dom {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉}) |
3 | fvex 6683 | . . 3 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
4 | dmsnopg 6070 | . . 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 2856 | 1 ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 〈cop 4573 class class class wbr 5066 dom cdm 5555 ↾ cres 5557 “ cima 5558 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 0cc0 10537 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Vtxcvtx 26781 iEdgciedg 26782 Trailsctrls 27472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-dm 5565 df-iota 6314 df-fv 6363 |
This theorem is referenced by: trlsegvdeglem7 28005 trlsegvdeg 28006 |
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