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Mirrors > Home > MPE Home > Th. List > curry1f | Structured version Visualization version GIF version |
Description: Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.) |
Ref | Expression |
---|---|
curry1.1 | ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) |
Ref | Expression |
---|---|
curry1f | ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6507 | . . 3 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵)) | |
2 | curry1.1 | . . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) | |
3 | 2 | curry1 7791 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
4 | 1, 3 | sylan 582 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
5 | fovrn 7310 | . . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷) | |
6 | 5 | 3expa 1113 | . 2 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷) |
7 | 4, 6 | fmpt3d 6873 | 1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 {csn 4559 ↦ cmpt 5137 × cxp 5546 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 (class class class)co 7148 2nd c2nd 7680 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-1st 7681 df-2nd 7682 |
This theorem is referenced by: nvinvfval 28409 |
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