Theorem ofs1 14893

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 766	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆)
4		simplr 768	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐵 ∈ 𝑇)
5		s1val 14522	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ⟨“𝐴”⟩ = {⟨0, 𝐴⟩})
6		0nn0 12416	. . . . . 6 ⊢ 0 ∈ ℕ0
7		fmptsn 7113	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {⟨0, 𝐴⟩} = (𝑖 ∈ {0} ↦ 𝐴))
8	6, 7	mpan 690	. . . . 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 14522	. . . . 5 ⊢ (𝐵 ∈ 𝑇 → ⟨“𝐵”⟩ = {⟨0, 𝐵⟩})
12		fmptsn 7113	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐵 ∈ 𝑇) → {⟨0, 𝐵⟩} = (𝑖 ∈ {0} ↦ 𝐵))
13	6, 12	mpan 690	. . . . 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 7642	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
17		ovex 7391	. . . 4 ⊢ (𝐴𝑅𝐵) ∈ V
18		s1val 14522	. . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩})
19	17, 18	ax-mp 5	. . 3 ⊢ ⟨“(𝐴𝑅𝐵)”⟩ = {⟨0, (𝐴𝑅𝐵)⟩}
20		fmptsn 7113	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {⟨0, (𝐴𝑅𝐵)⟩} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)))
21	6, 17, 20	mp2an 692	. . 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 1541   ∈ wcel 2113  Vcvv 3440  {csn 4580  ⟨cop 4586   ↦ cmpt 5179  (class class class)co 7358   ∘_f cof 7620  0cc0 11026  ℕ₀cn0 12401  ⟨“cs1 14519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-mulcl 11088  ax-i2m1 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-n0 12402  df-s1 14520

This theorem is referenced by:  ofs2  14894  1arithidomlem2  33617