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Theorem s1co 13378
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 13179 . . . . 5 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 0cn 9888 . . . . . 6 0 ∈ ℂ
3 xpsng 6296 . . . . . 6 ((0 ∈ ℂ ∧ 𝑆𝐴) → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
42, 3mpan 701 . . . . 5 (𝑆𝐴 → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
51, 4eqtr4d 2646 . . . 4 (𝑆𝐴 → ⟨“𝑆”⟩ = ({0} × {𝑆}))
65adantr 479 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“𝑆”⟩ = ({0} × {𝑆}))
76coeq2d 5193 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = (𝐹 ∘ ({0} × {𝑆})))
8 ffn 5943 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
9 id 22 . . . 4 (𝑆𝐴𝑆𝐴)
10 fcoconst 6291 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
118, 9, 10syl2anr 493 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
12 fvex 6097 . . . . 5 (𝐹𝑆) ∈ V
13 s1val 13179 . . . . 5 ((𝐹𝑆) ∈ V → ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩})
1412, 13ax-mp 5 . . . 4 ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩}
15 c0ex 9890 . . . . 5 0 ∈ V
1615, 12xpsn 6297 . . . 4 ({0} × {(𝐹𝑆)}) = {⟨0, (𝐹𝑆)⟩}
1714, 16eqtr4i 2634 . . 3 ⟨“(𝐹𝑆)”⟩ = ({0} × {(𝐹𝑆)})
1811, 17syl6reqr 2662 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“(𝐹𝑆)”⟩ = (𝐹 ∘ ({0} × {𝑆})))
197, 18eqtr4d 2646 1 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  {csn 4124  cop 4130   × cxp 5025  ccom 5031   Fn wfn 5784  wf 5785  cfv 5789  cc 9790  0cc0 9792  ⟨“cs1 13097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-mulcl 9854  ax-i2m1 9860
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-s1 13105
This theorem is referenced by:  cats1co  13400  s2co  13463  frmdgsum  17170  frmdup2  17173  efginvrel2  17911  vrgpinv  17953  frgpup2  17960  mrsubcv  30454
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