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Theorem ofcs1 32054
 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 5304 . . . 4 {0} ∈ V
21a1i 11 . . 3 ((𝐴𝑆𝐵𝑇) → {0} ∈ V)
3 simpr 488 . . 3 ((𝐴𝑆𝐵𝑇) → 𝐵𝑇)
4 simpll 766 . . 3 (((𝐴𝑆𝐵𝑇) ∧ 𝑖 ∈ {0}) → 𝐴𝑆)
5 s1val 14012 . . . . 5 (𝐴𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6 0nn0 11962 . . . . . 6 0 ∈ ℕ0
7 fmptsn 6926 . . . . . 6 ((0 ∈ ℕ0𝐴𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
86, 7mpan 689 . . . . 5 (𝐴𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
95, 8eqtrd 2793 . . . 4 (𝐴𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
109adantr 484 . . 3 ((𝐴𝑆𝐵𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
112, 3, 4, 10ofcfval2 31603 . 2 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
12 ovex 7189 . . . 4 (𝐴𝑅𝐵) ∈ V
13 s1val 14012 . . . 4 ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
1412, 13ax-mp 5 . . 3 ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
15 fmptsn 6926 . . . 4 ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
166, 12, 15mp2an 691 . . 3 {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1714, 16eqtri 2781 . 2 ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
1811, 17eqtr4di 2811 1 ((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f/c 𝑅𝐵) = ⟨“(𝐴𝑅𝐵)”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531   ↦ cmpt 5116  (class class class)co 7156  0cc0 10588  ℕ0cn0 11947  ⟨“cs1 14009   ∘f/c cofc 31594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-mulcl 10650  ax-i2m1 10656 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-n0 11948  df-s1 14010  df-ofc 31595 This theorem is referenced by:  ofcs2  32055
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