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Theorem ofcs1 31713
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
Assertion
Ref Expression
ofcs1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)

Proof of Theorem ofcs1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 snex 5322 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpr 485 . . 3 ((𝐴𝑆𝐵𝑇) → 𝐵𝑇)
4 simpll 763 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
5 s1val 13940 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 11900 . . . . . 6 0 ∈ ℕ0
7 fmptsn 6921 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 686 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2853 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 481 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
112, 3, 4, 10ofcfval2 31262 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
12 ovex 7178 . . . 4 (𝐴𝑅𝐵) ∈ V
13 s1val 13940 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1412, 13ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
15 fmptsn 6921 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
166, 12, 15mp2an 688 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1714, 16eqtri 2841 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1811, 17syl6eqr 2871 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563  cmpt 5137  (class class class)co 7145  0cc0 10525  0cn0 11885  ⟨“cs1 13937  f/c cofc 31253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-n0 11886  df-s1 13938  df-ofc 31254
This theorem is referenced by:  ofcs2  31714
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