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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 7395 | . . 3 ⊢ (𝜑 → curry 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶))) |
5 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑎(𝑦 ∈ 𝑌 ↦ 𝐶) | |
6 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥𝑌 | |
7 | nfcsb1v 3549 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 | |
8 | 6, 7 | nfmpt 4746 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶) |
9 | csbeq1a 3542 | . . . . 5 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶) | |
10 | 9 | mpteq2dv 4745 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
11 | 5, 8, 10 | cbvmpt 4749 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶)) |
12 | 4, 11 | syl6eq 2672 | . 2 ⊢ (𝜑 → curry 𝐹 = (𝑎 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⦋𝑎 / 𝑥⦌𝐶))) |
13 | csbeq1 3536 | . . . 4 ⊢ (𝑎 = 𝐴 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
14 | 13 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶) |
15 | 14 | mpteq2dv 4745 | . 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 6484 | . . 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 6288 | 1 ⊢ (𝜑 → (curry 𝐹‘𝐴) = (𝑦 ∈ 𝑌 ↦ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 Vcvv 3200 ⦋csb 3533 ∅c0 3915 ↦ cmpt 4729 ‘cfv 5888 ↦ cmpt2 6652 curry ccur 7391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-cur 7393 |
This theorem is referenced by: fvmpt2curryd 7397 pmatcollpw3lem 20588 logbmpt 24526 |
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