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Theorem s1co 13430
 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 13231 . . . . 5 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 0cn 9911 . . . . . 6 0 ∈ ℂ
3 xpsng 6312 . . . . . 6 ((0 ∈ ℂ ∧ 𝑆𝐴) → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
42, 3mpan 702 . . . . 5 (𝑆𝐴 → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
51, 4eqtr4d 2647 . . . 4 (𝑆𝐴 → ⟨“𝑆”⟩ = ({0} × {𝑆}))
65adantr 480 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“𝑆”⟩ = ({0} × {𝑆}))
76coeq2d 5206 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = (𝐹 ∘ ({0} × {𝑆})))
8 ffn 5958 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
9 id 22 . . . 4 (𝑆𝐴𝑆𝐴)
10 fcoconst 6307 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
118, 9, 10syl2anr 494 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
12 fvex 6113 . . . . 5 (𝐹𝑆) ∈ V
13 s1val 13231 . . . . 5 ((𝐹𝑆) ∈ V → ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩})
1412, 13ax-mp 5 . . . 4 ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩}
15 c0ex 9913 . . . . 5 0 ∈ V
1615, 12xpsn 6313 . . . 4 ({0} × {(𝐹𝑆)}) = {⟨0, (𝐹𝑆)⟩}
1714, 16eqtr4i 2635 . . 3 ⟨“(𝐹𝑆)”⟩ = ({0} × {(𝐹𝑆)})
1811, 17syl6reqr 2663 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“(𝐹𝑆)”⟩ = (𝐹 ∘ ({0} × {𝑆})))
197, 18eqtr4d 2647 1 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   × cxp 5036   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  ℂcc 9813  0cc0 9815  ⟨“cs1 13149 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-s1 13157 This theorem is referenced by:  cats1co  13452  s2co  13515  frmdgsum  17222  frmdup2  17225  efginvrel2  17963  vrgpinv  18005  frgpup2  18012  mrsubcv  30661
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