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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 5442 | . . . 4 ⊢ {0} ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V) |
3 | simpr 484 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 𝐵 ∈ 𝑇) | |
4 | simpll 767 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆) | |
5 | s1val 14633 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 0nn0 12539 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | fmptsn 7187 | . . . . . 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 2775 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
10 | 9 | adantr 480 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
11 | 2, 3, 4, 10 | ofcfval2 34085 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) |
12 | ovex 7464 | . . . 4 ⊢ (𝐴𝑅𝐵) ∈ V | |
13 | s1val 14633 | . . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉}) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉} |
15 | fmptsn 7187 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) | |
16 | 6, 12, 15 | mp2an 692 | . . 3 ⊢ {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
17 | 14, 16 | eqtri 2763 | . 2 ⊢ 〈“(𝐴𝑅𝐵)”〉 = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
18 | 11, 17 | eqtr4di 2793 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = 〈“(𝐴𝑅𝐵)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ↦ cmpt 5231 (class class class)co 7431 0cc0 11153 ℕ0cn0 12524 〈“cs1 14630 ∘f/c cofc 34076 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-n0 12525 df-s1 14631 df-ofc 34077 |
This theorem is referenced by: ofcs2 34539 |
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