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Mirrors > Home > MPE Home > Th. List > curry2f | Structured version Visualization version GIF version |
Description: Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.) |
Ref | Expression |
---|---|
curry2.1 | ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) |
Ref | Expression |
---|---|
curry2f | ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovrn 6702 | . . . . 5 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝑥𝐹𝐶) ∈ 𝐷) | |
2 | 1 | 3com23 1263 | . . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷) |
3 | 2 | 3expa 1257 | . . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷) |
4 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) | |
5 | 3, 4 | fmptd 6292 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)):𝐴⟶𝐷) |
6 | ffn 5958 | . . . 4 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵)) | |
7 | curry2.1 | . . . . 5 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) | |
8 | 7 | curry2 7159 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
9 | 6, 8 | sylan 487 | . . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
10 | 9 | feq1d 5943 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → (𝐺:𝐴⟶𝐷 ↔ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)):𝐴⟶𝐷)) |
11 | 5, 10 | mpbird 246 | 1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 (class class class)co 6549 1st c1st 7057 |
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-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-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-1st 7059 df-2nd 7060 |
This theorem is referenced by: curry2ima 28869 |
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