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Mirrors > Home > MPE Home > Th. List > mpocurryvald | Structured version Visualization version GIF version |
Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.) |
Ref | Expression |
---|---|
mpocurryd.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
mpocurryd.c | ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) |
mpocurryd.n | ⊢ (𝜑 → 𝑌 ≠ ∅) |
mpocurryvald.y | ⊢ (𝜑 → 𝑌 ∈ 𝑊) |
mpocurryvald.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Ref | Expression |
---|---|
mpocurryvald | ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpocurryd.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) | |
2 | mpocurryd.c | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) | |
3 | mpocurryd.n | . . . 4 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
4 | 1, 2, 3 | mpocurryd 7918 | . . 3 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) |
5 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑎(𝑦 ∈ 𝑌 ↦ 𝐶) | |
6 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑌 | |
7 | nfcsb1v 3852 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 | |
8 | 6, 7 | nfmpt 5127 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) |
9 | csbeq1a 3842 | . . . . 5 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶) | |
10 | 9 | mpteq2dv 5126 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
11 | 5, 8, 10 | cbvmpt 5131 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
12 | 4, 11 | eqtrdi 2849 | . 2 ⊢ (𝜑 → curry 𝐹 = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))) |
13 | csbeq1 3831 | . . . 4 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
14 | 13 | adantl 485 | . . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) |
15 | 14 | mpteq2dv 5126 | . 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
16 | mpocurryvald.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
17 | mpocurryvald.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑊) | |
18 | 17 | mptexd 6964 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶) ∈ V) |
19 | 12, 15, 16, 18 | fvmptd 6752 | 1 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 ⦋csb 3828 ∅c0 4243 ↦ cmpt 5110 ‘cfv 6324 ∈ cmpo 7137 curry ccur 7914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-cur 7916 |
This theorem is referenced by: fvmpocurryd 7920 pmatcollpw3lem 21388 logbmpt 25374 |
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