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Mirrors > Home > MPE Home > Th. List > mpt2curryvald | 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 |
---|---|
mpt2curryd.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) |
mpt2curryd.c | ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) |
mpt2curryd.n | ⊢ (𝜑 → 𝑌 ≠ ∅) |
mpt2curryvald.y | ⊢ (𝜑 → 𝑌 ∈ 𝑊) |
mpt2curryvald.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Ref | Expression |
---|---|
mpt2curryvald | ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2curryd.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐶) | |
2 | mpt2curryd.c | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐶 ∈ 𝑉) | |
3 | mpt2curryd.n | . . . 4 ⊢ (𝜑 → 𝑌 ≠ ∅) | |
4 | 1, 2, 3 | mpt2curryd 7282 | . . 3 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) |
5 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑎(𝑦 ∈ 𝑌 ↦ 𝐶) | |
6 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑌 | |
7 | nfcsb1v 3515 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 | |
8 | 6, 7 | nfmpt 4674 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) |
9 | csbeq1a 3508 | . . . . 5 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶) | |
10 | 9 | mpteq2dv 4673 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
11 | 5, 8, 10 | cbvmpt 4677 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
12 | 4, 11 | syl6eq 2660 | . 2 ⊢ (𝜑 → curry 𝐹 = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))) |
13 | csbeq1 3502 | . . . 4 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
14 | 13 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) |
15 | 14 | mpteq2dv 4673 | . 2 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
16 | mpt2curryvald.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
17 | mpt2curryvald.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑊) | |
18 | mptexg 6389 | . . 3 ⊢ (𝑌 ∈ 𝑊 → (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶) ∈ V) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶) ∈ V) |
20 | 12, 15, 16, 19 | fvmptd 6197 | 1 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 Vcvv 3173 ⦋csb 3499 ∅c0 3874 ↦ cmpt 4643 ‘cfv 5804 ↦ cmpt2 6551 curry ccur 7278 |
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-un 6847 |
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-1st 7059 df-2nd 7060 df-cur 7280 |
This theorem is referenced by: fvmpt2curryd 7284 pmatcollpw3lem 20407 logbmpt 24326 |
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