Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > trlsegvdeglem3 | Structured version Visualization version GIF version |
Description: Lemma for trlsegvdeg 27087. (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 6201 | . . . 4 ⊢ (𝐹‘𝑁) ∈ V | |
2 | fvex 6201 | . . . 4 ⊢ (𝐼‘(𝐹‘𝑁)) ∈ V | |
3 | 1, 2 | pm3.2i 471 | . . 3 ⊢ ((𝐹‘𝑁) ∈ V ∧ (𝐼‘(𝐹‘𝑁)) ∈ V) |
4 | funsng 5937 | . . 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 5910 | . 2 ⊢ (𝜑 → (Fun (iEdg‘𝑌) ↔ Fun {〈(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))〉})) |
8 | 5, 7 | mpbird 247 | 1 ⊢ (𝜑 → Fun (iEdg‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 class class class wbr 4653 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ...cfz 12326 ..^cfzo 12465 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 Trailsctrls 26587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: trlsegvdeg 27087 |
Copyright terms: Public domain | W3C validator |