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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 | fovrn 6971 | . . . 4 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷) | |
2 | 1 | 3expa 1112 | . . 3 ⊢ (((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐶𝐹𝑥) ∈ 𝐷) |
3 | eqid 2761 | . . 3 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) | |
4 | 2, 3 | fmptd 6550 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)):𝐵⟶𝐷) |
5 | ffn 6207 | . . . 4 ⊢ (𝐹:(𝐴 × 𝐵)⟶𝐷 → 𝐹 Fn (𝐴 × 𝐵)) | |
6 | curry1.1 | . . . . 5 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) | |
7 | 6 | curry1 7439 | . . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
8 | 5, 7 | sylan 489 | . . 3 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
9 | 8 | feq1d 6192 | . 2 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → (𝐺:𝐵⟶𝐷 ↔ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)):𝐵⟶𝐷)) |
10 | 4, 9 | mpbird 247 | 1 ⊢ ((𝐹:(𝐴 × 𝐵)⟶𝐷 ∧ 𝐶 ∈ 𝐴) → 𝐺:𝐵⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2140 Vcvv 3341 {csn 4322 ↦ cmpt 4882 × cxp 5265 ◡ccnv 5266 ↾ cres 5269 ∘ ccom 5271 Fn wfn 6045 ⟶wf 6046 (class class class)co 6815 2nd c2nd 7334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2142 ax-9 2149 ax-10 2169 ax-11 2184 ax-12 2197 ax-13 2392 ax-ext 2741 ax-sep 4934 ax-nul 4942 ax-pow 4993 ax-pr 5056 ax-un 7116 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2048 df-eu 2612 df-mo 2613 df-clab 2748 df-cleq 2754 df-clel 2757 df-nfc 2892 df-ne 2934 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3343 df-sbc 3578 df-csb 3676 df-dif 3719 df-un 3721 df-in 3723 df-ss 3730 df-nul 4060 df-if 4232 df-sn 4323 df-pr 4325 df-op 4329 df-uni 4590 df-iun 4675 df-br 4806 df-opab 4866 df-mpt 4883 df-id 5175 df-xp 5273 df-rel 5274 df-cnv 5275 df-co 5276 df-dm 5277 df-rn 5278 df-res 5279 df-ima 5280 df-iota 6013 df-fun 6052 df-fn 6053 df-f 6054 df-f1 6055 df-fo 6056 df-f1o 6057 df-fv 6058 df-ov 6818 df-1st 7335 df-2nd 7336 |
This theorem is referenced by: nvinvfval 27826 |
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