Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > curry2val | Structured version Visualization version GIF version |
Description: The value 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 |
---|---|
curry2val | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | curry2.1 | . . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(1st ↾ (V × {𝐶}))) | |
2 | 1 | curry2 7805 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))) |
3 | 2 | fveq1d 6675 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷)) |
4 | eqid 2824 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶)) | |
5 | 4 | fvmptndm 6801 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅) |
6 | 5 | adantl 484 | . . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = ∅) |
7 | fndm 6458 | . . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) | |
8 | simpl 485 | . . . . . . . 8 ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐷 ∈ 𝐴) | |
9 | 8 | con3i 157 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐴 → ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) |
10 | ndmovg 7334 | . . . . . . 7 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)) → (𝐷𝐹𝐶) = ∅) | |
11 | 7, 9, 10 | syl2an 597 | . . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → (𝐷𝐹𝐶) = ∅) |
12 | 6, 11 | eqtr4d 2862 | . . . . 5 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ¬ 𝐷 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)) |
13 | 12 | ex 415 | . . . 4 ⊢ (𝐹 Fn (𝐴 × 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))) |
14 | 13 | adantr 483 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (¬ 𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶))) |
15 | oveq1 7166 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝑥𝐹𝐶) = (𝐷𝐹𝐶)) | |
16 | ovex 7192 | . . . 4 ⊢ (𝐷𝐹𝐶) ∈ V | |
17 | 15, 4, 16 | fvmpt 6771 | . . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)) |
18 | 14, 17 | pm2.61d2 183 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))‘𝐷) = (𝐷𝐹𝐶)) |
19 | 3, 18 | eqtrd 2859 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝐺‘𝐷) = (𝐷𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 {csn 4570 ↦ cmpt 5149 × cxp 5556 ◡ccnv 5557 dom cdm 5558 ↾ cres 5560 ∘ ccom 5562 Fn wfn 6353 ‘cfv 6358 (class class class)co 7159 1st c1st 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-1st 7692 df-2nd 7693 |
This theorem is referenced by: curry2ima 30447 |
Copyright terms: Public domain | W3C validator |