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Theorem s1co 13788
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 13578 . . . . 5 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 0cn 10234 . . . . . 6 0 ∈ ℂ
3 xpsng 6549 . . . . . 6 ((0 ∈ ℂ ∧ 𝑆𝐴) → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
42, 3mpan 670 . . . . 5 (𝑆𝐴 → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
51, 4eqtr4d 2808 . . . 4 (𝑆𝐴 → ⟨“𝑆”⟩ = ({0} × {𝑆}))
65adantr 466 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“𝑆”⟩ = ({0} × {𝑆}))
76coeq2d 5423 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = (𝐹 ∘ ({0} × {𝑆})))
8 ffn 6185 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
9 id 22 . . . 4 (𝑆𝐴𝑆𝐴)
10 fcoconst 6544 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
118, 9, 10syl2anr 584 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
12 fvex 6342 . . . . 5 (𝐹𝑆) ∈ V
13 s1val 13578 . . . . 5 ((𝐹𝑆) ∈ V → ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩})
1412, 13ax-mp 5 . . . 4 ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩}
15 c0ex 10236 . . . . 5 0 ∈ V
1615, 12xpsn 6550 . . . 4 ({0} × {(𝐹𝑆)}) = {⟨0, (𝐹𝑆)⟩}
1714, 16eqtr4i 2796 . . 3 ⟨“(𝐹𝑆)”⟩ = ({0} × {(𝐹𝑆)})
1811, 17syl6reqr 2824 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“(𝐹𝑆)”⟩ = (𝐹 ∘ ({0} × {𝑆})))
197, 18eqtr4d 2808 1 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322   × cxp 5247  ccom 5253   Fn wfn 6026  wf 6027  cfv 6031  cc 10136  0cc0 10138  ⟨“cs1 13490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-mulcl 10200  ax-i2m1 10206
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-s1 13498
This theorem is referenced by:  cats1co  13810  s2co  13874  frmdgsum  17607  frmdup2  17610  efginvrel2  18347  vrgpinv  18389  frgpup2  18396  mrsubcv  31745
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