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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 | ffn 6720 | . . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵)) | |
2 | curry2.1 | . . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) | |
3 | 2 | curry2 8113 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
4 | 1, 3 | sylan 578 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
5 | fovcdm 7588 | . . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝑥𝐹𝐶) ∈ 𝐷) | |
6 | 5 | 3com23 1123 | . . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷) |
7 | 6 | 3expa 1115 | . 2 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹𝐶) ∈ 𝐷) |
8 | 4, 7 | fmpt3d 7122 | 1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐵) → 𝐺:𝐴⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4623 ↦ cmpt 5228 × cxp 5672 ◡ccnv 5673 ↾ cres 5676 ∘ ccom 5678 Fn wfn 6541 ⟶wf 6542 (class class class)co 7416 1st c1st 7993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-1st 7995 df-2nd 7996 |
This theorem is referenced by: curry2ima 32620 |
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