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Mirrors > Home > MPE Home > Th. List > ofs1 | Structured version Visualization version GIF version |
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
Ref | Expression |
---|---|
ofs1 | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐵”〉) = 〈“(𝐴𝑅𝐵)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5323 | . . . 4 ⊢ {0} ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V) |
3 | simpll 765 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆) | |
4 | simplr 767 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐵 ∈ 𝑇) | |
5 | s1val 13946 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 0nn0 11906 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | fmptsn 6923 | . . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) | |
8 | 6, 7 | mpan 688 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) |
9 | 5, 8 | eqtrd 2856 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
11 | s1val 13946 | . . . . 5 ⊢ (𝐵 ∈ 𝑇 → 〈“𝐵”〉 = {〈0, 𝐵〉}) | |
12 | fmptsn 6923 | . . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐵 ∈ 𝑇) → {〈0, 𝐵〉} = (𝑖 ∈ {0} ↦ 𝐵)) | |
13 | 6, 12 | mpan 688 | . . . . 5 ⊢ (𝐵 ∈ 𝑇 → {〈0, 𝐵〉} = (𝑖 ∈ {0} ↦ 𝐵)) |
14 | 11, 13 | eqtrd 2856 | . . . 4 ⊢ (𝐵 ∈ 𝑇 → 〈“𝐵”〉 = (𝑖 ∈ {0} ↦ 𝐵)) |
15 | 14 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐵”〉 = (𝑖 ∈ {0} ↦ 𝐵)) |
16 | 2, 3, 4, 10, 15 | offval2 7420 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐵”〉) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) |
17 | ovex 7183 | . . . 4 ⊢ (𝐴𝑅𝐵) ∈ V | |
18 | s1val 13946 | . . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉}) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉} |
20 | fmptsn 6923 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) | |
21 | 6, 17, 20 | mp2an 690 | . . 3 ⊢ {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
22 | 19, 21 | eqtri 2844 | . 2 ⊢ 〈“(𝐴𝑅𝐵)”〉 = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
23 | 16, 22 | syl6eqr 2874 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐵”〉) = 〈“(𝐴𝑅𝐵)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 ↦ cmpt 5138 (class class class)co 7150 ∘f cof 7401 0cc0 10531 ℕ0cn0 11891 〈“cs1 13943 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-mulcl 10593 ax-i2m1 10599 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-n0 11892 df-s1 13944 |
This theorem is referenced by: ofs2 14325 |
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