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Theorem ofs1 13638
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 4874 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpll 789 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
4 simplr 791 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐵𝑇)
5 s1val 13312 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 11252 . . . . . 6 0 ∈ ℕ0
7 fmptsn 6388 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 705 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2660 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 481 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
11 s1val 13312 . . . . 5 (𝐵𝑇 → ⟨“𝐵”⟩ = {⟨0, 𝐵⟩})
12 fmptsn 6388 . . . . . 6 ((0 ∈ ℕ0𝐵𝑇) → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
136, 12mpan 705 . . . . 5 (𝐵𝑇 → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
1411, 13eqtrd 2660 . . . 4 (𝐵𝑇 → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
1514adantl 482 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
162, 3, 4, 10, 15offval2 6868 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
17 ovex 6633 . . . 4 (𝐴𝑅𝐵) ∈ V
18 s1val 13312 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1917, 18ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
20 fmptsn 6388 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
216, 17, 20mp2an 707 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
2219, 21eqtri 2648 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
2316, 22syl6eqr 2678 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cop 4159  cmpt 4678  (class class class)co 6605  𝑓 cof 6849  0cc0 9881  0cn0 11237  ⟨“cs1 13228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-mulcl 9943  ax-i2m1 9949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-n0 11238  df-s1 13236
This theorem is referenced by:  ofs2  13639
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