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 5335 | . . . 4 ⊢ {0} ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → {0} ∈ V) |
3 | simpr 487 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 𝐵 ∈ 𝑇) | |
4 | simpll 765 | . . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) ∧ 𝑖 ∈ {0}) → 𝐴 ∈ 𝑆) | |
5 | s1val 13955 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = {〈0, 𝐴〉}) | |
6 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | fmptsn 6932 | . . . . . 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 2859 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
10 | 9 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → 〈“𝐴”〉 = (𝑖 ∈ {0} ↦ 𝐴)) |
11 | 2, 3, 4, 10 | ofcfval2 31367 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) |
12 | ovex 7192 | . . . 4 ⊢ (𝐴𝑅𝐵) ∈ V | |
13 | s1val 13955 | . . . 4 ⊢ ((𝐴𝑅𝐵) ∈ V → 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉}) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ 〈“(𝐴𝑅𝐵)”〉 = {〈0, (𝐴𝑅𝐵)〉} |
15 | fmptsn 6932 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ (𝐴𝑅𝐵) ∈ V) → {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵))) | |
16 | 6, 12, 15 | mp2an 690 | . . 3 ⊢ {〈0, (𝐴𝑅𝐵)〉} = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
17 | 14, 16 | eqtri 2847 | . 2 ⊢ 〈“(𝐴𝑅𝐵)”〉 = (𝑖 ∈ {0} ↦ (𝐴𝑅𝐵)) |
18 | 11, 17 | syl6eqr 2877 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f/c 𝑅𝐵) = 〈“(𝐴𝑅𝐵)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 ↦ cmpt 5149 (class class class)co 7159 0cc0 10540 ℕ0cn0 11900 〈“cs1 13952 ∘f/c cofc 31358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-i2m1 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-n0 11901 df-s1 13953 df-ofc 31359 |
This theorem is referenced by: ofcs2 31819 |
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