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Theorem ofcs1 34176
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 5433 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpr 484 . . 3 ((𝐴𝑆𝐵𝑇) → 𝐵𝑇)
4 simpll 766 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
5 s1val 14581 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 12518 . . . . . 6 0 ∈ ℕ0
7 fmptsn 7176 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 689 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2768 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 480 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
112, 3, 4, 10ofcfval2 33723 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
12 ovex 7453 . . . 4 (𝐴𝑅𝐵) ∈ V
13 s1val 14581 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1412, 13ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
15 fmptsn 7176 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
166, 12, 15mp2an 691 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1714, 16eqtri 2756 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1811, 17eqtr4di 2786 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  {csn 4629  cop 4635  cmpt 5231  (class class class)co 7420  0cc0 11139  0cn0 12503  ⟨“cs1 14578  f/c cofc 33714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-mulcl 11201  ax-i2m1 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-n0 12504  df-s1 14579  df-ofc 33715
This theorem is referenced by:  ofcs2  34177
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