Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofcs1 Structured version   Visualization version   GIF version

Theorem ofcs1 34538
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 5442 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpr 484 . . 3 ((𝐴𝑆𝐵𝑇) → 𝐵𝑇)
4 simpll 767 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
5 s1val 14633 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 12539 . . . . . 6 0 ∈ ℕ0
7 fmptsn 7187 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 690 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2775 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 480 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
112, 3, 4, 10ofcfval2 34085 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
12 ovex 7464 . . . 4 (𝐴𝑅𝐵) ∈ V
13 s1val 14633 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1412, 13ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
15 fmptsn 7187 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
166, 12, 15mp2an 692 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1714, 16eqtri 2763 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1811, 17eqtr4di 2793 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637  cmpt 5231  (class class class)co 7431  0cc0 11153  0cn0 12524  ⟨“cs1 14630  f/c cofc 34076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-mulcl 11215  ax-i2m1 11221
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-n0 12525  df-s1 14631  df-ofc 34077
This theorem is referenced by:  ofcs2  34539
  Copyright terms: Public domain W3C validator