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Theorem s1co 14190
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 13947 . . . . 5 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 0cn 10626 . . . . . 6 0 ∈ ℂ
3 xpsng 6882 . . . . . 6 ((0 ∈ ℂ ∧ 𝑆𝐴) → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
42, 3mpan 689 . . . . 5 (𝑆𝐴 → ({0} × {𝑆}) = {⟨0, 𝑆⟩})
51, 4eqtr4d 2839 . . . 4 (𝑆𝐴 → ⟨“𝑆”⟩ = ({0} × {𝑆}))
65adantr 484 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“𝑆”⟩ = ({0} × {𝑆}))
76coeq2d 5701 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = (𝐹 ∘ ({0} × {𝑆})))
8 fvex 6662 . . . . 5 (𝐹𝑆) ∈ V
9 s1val 13947 . . . . 5 ((𝐹𝑆) ∈ V → ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩})
108, 9ax-mp 5 . . . 4 ⟨“(𝐹𝑆)”⟩ = {⟨0, (𝐹𝑆)⟩}
11 c0ex 10628 . . . . 5 0 ∈ V
1211, 8xpsn 6884 . . . 4 ({0} × {(𝐹𝑆)}) = {⟨0, (𝐹𝑆)⟩}
1310, 12eqtr4i 2827 . . 3 ⟨“(𝐹𝑆)”⟩ = ({0} × {(𝐹𝑆)})
14 ffn 6491 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
15 id 22 . . . 4 (𝑆𝐴𝑆𝐴)
16 fcoconst 6877 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
1714, 15, 16syl2anr 599 . . 3 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹𝑆)}))
1813, 17eqtr4id 2855 . 2 ((𝑆𝐴𝐹:𝐴𝐵) → ⟨“(𝐹𝑆)”⟩ = (𝐹 ∘ ({0} × {𝑆})))
197, 18eqtr4d 2839 1 ((𝑆𝐴𝐹:𝐴𝐵) → (𝐹 ∘ ⟨“𝑆”⟩) = ⟨“(𝐹𝑆)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  {csn 4528  cop 4534   × cxp 5521  ccom 5527   Fn wfn 6323  wf 6324  cfv 6328  cc 10528  0cc0 10530  ⟨“cs1 13944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-mulcl 10592  ax-i2m1 10598
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-s1 13945
This theorem is referenced by:  cats1co  14213  s2co  14277  frmdgsum  18022  frmdup2  18025  efginvrel2  18848  vrgpinv  18890  frgpup2  18897  mrsubcv  32865
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