MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s1co Structured version   Visualization version   GIF version

Theorem s1co 14019
Description: Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1co ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)

Proof of Theorem s1co
StepHypRef Expression
1 s1val 13784 . . . . 5 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 0cn 10468 . . . . . 6 0 ∈ ℂ
3 xpsng 6755 . . . . . 6 ((0 ∈ ℂ ∧ 𝑆𝐴) → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
42, 3mpan 686 . . . . 5 (𝑆𝐴 → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
51, 4eqtr4d 2832 . . . 4 (𝑆𝐴 → ⟨“𝑆”⟩ = ({0} × {𝑆}))
65adantr 481 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“𝑆”⟩ = ({0} × {𝑆}))
76coeq2d 5611 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = (𝐹 ∘ ({0} × {𝑆})))
8 ffn 6374 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
9 id 22 . . . 4 (𝑆𝐴𝑆𝐴)
10 fcoconst 6750 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
118, 9, 10syl2anr 596 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
12 fvex 6543 . . . . 5 (𝐹𝑆) ∈ V
13 s1val 13784 . . . . 5 ((𝐹𝑆) ∈ V → ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩})
1412, 13ax-mp 5 . . . 4 ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩}
15 c0ex 10470 . . . . 5 0 ∈ V
1615, 12xpsn 6757 . . . 4 ({0} × {(𝐹𝑆)}) = {⟨0, (𝐹𝑆)⟩}
1714, 16eqtr4i 2820 . . 3 ⟨“(𝐹𝑆)”⟩ = ({0} × {(𝐹𝑆)})
1811, 17syl6reqr 2848 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“(𝐹𝑆)”⟩ = (𝐹 ∘ ({0} × {𝑆})))
197, 18eqtr4d 2832 1 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  Vcvv 3432  {csn 4466  cop 4472   × cxp 5433  ccom 5439   Fn wfn 6212  wf 6213  cfv 6217  cc 10370  0cc0 10372  ⟨“cs1 13781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-mulcl 10434  ax-i2m1 10440
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-s1 13782
This theorem is referenced by:  cats1co  14042  s2co  14106  frmdgsum  17826  frmdup2  17829  efginvrel2  18568  vrgpinv  18610  frgpup2  18617  mrsubcv  32310
  Copyright terms: Public domain W3C validator