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Theorem ofs1 13910
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
Assertion
Ref Expression
ofs1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Proof of Theorem ofs1
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 snex 5057 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpll 807 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
4 simplr 809 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐵𝑇)
5 s1val 13568 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 11499 . . . . . 6 0 ∈ ℕ0
7 fmptsn 6597 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 708 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2794 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 472 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
11 s1val 13568 . . . . 5 (𝐵𝑇 → ⟨“𝐵”⟩ = {⟨0, 𝐵⟩})
12 fmptsn 6597 . . . . . 6 ((0 ∈ ℕ0𝐵𝑇) → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
136, 12mpan 708 . . . . 5 (𝐵𝑇 → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
1411, 13eqtrd 2794 . . . 4 (𝐵𝑇 → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
1514adantl 473 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
162, 3, 4, 10, 15offval2 7079 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
17 ovex 6841 . . . 4 (𝐴𝑅𝐵) ∈ V
18 s1val 13568 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1917, 18ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
20 fmptsn 6597 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
216, 17, 20mp2an 710 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
2219, 21eqtri 2782 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
2316, 22syl6eqr 2812 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327  cmpt 4881  (class class class)co 6813  𝑓 cof 7060  0cc0 10128  0cn0 11484  ⟨“cs1 13480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-mulcl 10190  ax-i2m1 10196
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-n0 11485  df-s1 13488
This theorem is referenced by:  ofs2  13911
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