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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 4835 | . . . 4 ⊢ {0} ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V) |
3 | simpll 786 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆) | |
4 | simplr 788 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐵 ∈ 𝑇) | |
5 | s1val 13231 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 0nn0 11184 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | fmptsn 6338 | . . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑆) → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) | |
8 | 6, 7 | mpan 702 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → {〈0, 𝐴〉} = (𝑖 ∈ {0} ↦ 𝐴)) |
9 | 5, 8 | eqtrd 2644 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
11 | s1val 13231 | . . . . 5 ⊢ (𝐵 ∈ 𝑇 → 〈“𝐵”〉 = {〈0, 𝐵〉}) | |
12 | fmptsn 6338 | . . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐵 ∈ 𝑇) → {〈0, 𝐵〉} = (𝑖 ∈ {0} ↦ 𝐵)) | |
13 | 6, 12 | mpan 702 | . . . . 5 ⊢ (𝐵 ∈ 𝑇 → {〈0, 𝐵〉} = (𝑖 ∈ {0} ↦ 𝐵)) |
14 | 11, 13 | eqtrd 2644 | . . . 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 6812 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘𝑓 𝑅〈“𝐵”〉) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) |
17 | ovex 6577 | . . . 4 ⊢ (𝐴𝑅𝐵) ∈ V | |
18 | s1val 13231 | . . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉}) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉} |
20 | fmptsn 6338 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) | |
21 | 6, 17, 20 | mp2an 704 | . . 3 ⊢ {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
22 | 19, 21 | eqtri 2632 | . 2 ⊢ 〈“(𝐴𝑅𝐵)”〉 = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
23 | 16, 22 | syl6eqr 2662 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘𝑓 𝑅〈“𝐵”〉) = 〈“(𝐴𝑅𝐵)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ↦ cmpt 4643 (class class class)co 6549 ∘𝑓 cof 6793 0cc0 9815 ℕ0cn0 11169 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-n0 11170 df-s1 13157 |
This theorem is referenced by: ofs2 13558 |
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