Theorem ofs1 14932

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 5381	. . . 4 ⊢ {0} ∈ V
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V)
3		simpll 767	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆)
4		simplr 769	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐵 ∈ 𝑇)
5		s1val 14561	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6		0nn0 12452	. . . . . 6 ⊢ 0 ∈ ℕ0
7		fmptsn 7122	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
8	6, 7	mpan 691	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
9	5, 8	eqtrd 2771	. . . 4 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ⟨“𝐴”⟩ = (𝑖 ∈ {0} ↦ 𝐴))
11		s1val 14561	. . . . 5 ⊢ (𝐵 ∈ 𝑇 → ⟨“𝐵”⟩ = {⟨0, 𝐵⟩})
12		fmptsn 7122	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐵 ∈ 𝑇) → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
13	6, 12	mpan 691	. . . . 5 ⊢ (𝐵 ∈ 𝑇 → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
14	11, 13	eqtrd 2771	. . . 4 ⊢ (𝐵 ∈ 𝑇 → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
15	14	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ⟨“𝐵”⟩ = (𝑖 ∈ {0} ↦ 𝐵))
16	2, 3, 4, 10, 15	offval2 7651	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
17		ovex 7400	. . . 4 ⊢ (𝐴𝑅𝐵) ∈ V
18		s1val 14561	. . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
19	17, 18	ax-mp 5	. . 3 ⊢ ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
20		fmptsn 7122	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
21	6, 17, 20	mp2an 693	. . 3 ⊢ {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
22	19, 21	eqtri 2759	. 2 ⊢ ⟨“(𝐴𝑅𝐵)”⟩ = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))
23	16, 22	eqtr4di 2789	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ⟨cop 4573   ↦ cmpt 5166  (class class class)co 7367   ∘_f cof 7629  0cc0 11038  ℕ₀cn0 12437  ⟨“cs1 14558

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-mulcl 11100  ax-i2m1 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-n0 12438  df-s1 14559

This theorem is referenced by:  ofs2  14933  1arithidomlem2  33596