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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 8204 | . . 3 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) |
5 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑎(𝑦 ∈ 𝑌 ↦ 𝐶) | |
6 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑌 | |
7 | nfcsb1v 3884 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 | |
8 | 6, 7 | nfmpt 5216 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) |
9 | csbeq1a 3873 | . . . . 5 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶) | |
10 | 9 | mpteq2dv 5211 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
11 | 5, 8, 10 | cbvmpt 5220 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
12 | 4, 11 | eqtrdi 2789 | . 2 ⊢ (𝜑 → curry 𝐹 = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))) |
13 | csbeq1 3862 | . . . 4 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
14 | 13 | adantl 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) |
15 | 14 | mpteq2dv 5211 | . 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 7178 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶) ∈ V) |
19 | 12, 15, 16, 18 | fvmptd 6959 | 1 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 Vcvv 3447 ⦋csb 3859 ∅c0 4286 ↦ cmpt 5192 ‘cfv 6500 ∈ cmpo 7363 curry ccur 8200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-cur 8202 |
This theorem is referenced by: fvmpocurryd 8206 pmatcollpw3lem 22155 logbmpt 26161 |
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