Theorem ofs1 14930

Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)

Assertion

Ref	Expression
ofs1	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Proof of Theorem ofs1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5375	. . . 4 ⊢ {0} ∈ V
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V)
3		simpll 772	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆)
4		simplr 774	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐵 ∈ 𝑇)
5		s1val 14559	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6		0nn0 12450	. . . . . 6 ⊢ 0 ∈ ℕ0
7		fmptsn 7118	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
8	6, 7	mpan 696	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
9	5, 8	eqtrd 2775	. . . 4 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
10	9	adantr 481	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
11		s1val 14559	. . . . 5 ⊢ (𝐵 ∈ 𝑇 → ⟨“𝐵”⟩ = {⟨0, 𝐵⟩})
12		fmptsn 7118	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐵 ∈ 𝑇) → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
13	6, 12	mpan 696	. . . . 5 ⊢ (𝐵 ∈ 𝑇 → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
14	11, 13	eqtrd 2775	. . . 4 ⊢ (𝐵 ∈ 𝑇 → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
15	14	adantl 482	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
16	2, 3, 4, 10, 15	offval2 7647	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
17		ovex 7396	. . . 4 ⊢ (𝐴𝑅𝐵) ∈ V
18		s1val 14559	. . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
19	17, 18	ax-mp 5	. . 3 ⊢ ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
20		fmptsn 7118	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
21	6, 17, 20	mp2an 698	. . 3 ⊢ {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
22	19, 21	eqtri 2763	. 2 ⊢ ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
23	16, 22	eqtr4di 2793	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562  ⟨cop 4568   ↦ cmpt 5160  (class class class)co 7363   ∘_f cof 7625  0cc0 11036  ℕ₀cn0 12435  ⟨“cs1 14556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-mulcl 11098  ax-i2m1 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-n0 12436  df-s1 14557

This theorem is referenced by:  ofs2  14931  1arithidomlem2  33626