Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcs1 | Structured version Visualization version GIF version |
Description: Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
Ref | Expression |
---|---|
ofcs1 | ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = 〈“(𝐴𝑅𝐵)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5322 | . . . 4 ⊢ {0} ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V) |
3 | simpr 485 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 𝐵 ∈ 𝑇) | |
4 | simpll 763 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆) | |
5 | s1val 13940 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 0nn0 11900 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | fmptsn 6921 | . . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) | |
8 | 6, 7 | mpan 686 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) |
9 | 5, 8 | eqtrd 2853 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
11 | 2, 3, 4, 10 | ofcfval2 31262 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) |
12 | ovex 7178 | . . . 4 ⊢ (𝐴𝑅𝐵) ∈ V | |
13 | s1val 13940 | . . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉}) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉} |
15 | fmptsn 6921 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) | |
16 | 6, 12, 15 | mp2an 688 | . . 3 ⊢ {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
17 | 14, 16 | eqtri 2841 | . 2 ⊢ 〈“(𝐴𝑅𝐵)”〉 = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
18 | 11, 17 | syl6eqr 2871 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = 〈“(𝐴𝑅𝐵)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ↦ cmpt 5137 (class class class)co 7145 0cc0 10525 ℕ0cn0 11885 〈“cs1 13937 ∘f/c cofc 31253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-n0 11886 df-s1 13938 df-ofc 31254 |
This theorem is referenced by: ofcs2 31714 |
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