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Mirrors > Home > MPE Home > Th. List > curry1val | Structured version Visualization version GIF version |
Description: The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
curry1.1 | ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) |
Ref | Expression |
---|---|
curry1val | ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | curry1.1 | . . . 4 ⊢ 𝐺 = (𝐹 ∘ ◡(2nd ↾ ({𝐶} × V))) | |
2 | 1 | curry1 7269 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
3 | 2 | fveq1d 6193 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷)) |
4 | eqid 2622 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) | |
5 | 4 | dmmptss 5631 | . . . . . . . . 9 ⊢ dom (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) ⊆ 𝐵 |
6 | 5 | sseli 3599 | . . . . . . . 8 ⊢ (𝐷 ∈ dom (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) → 𝐷 ∈ 𝐵) |
7 | 6 | con3i 150 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ 𝐷 ∈ dom (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))) |
8 | ndmfv 6218 | . . . . . . 7 ⊢ (¬ 𝐷 ∈ dom (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥)) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅) | |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅) |
10 | 9 | adantl 482 | . . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = ∅) |
11 | fndm 5990 | . . . . . . 7 ⊢ (𝐹 Fn (𝐴 × 𝐵) → dom 𝐹 = (𝐴 × 𝐵)) | |
12 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → dom 𝐹 = (𝐴 × 𝐵)) |
13 | simpr 477 | . . . . . . 7 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → 𝐷 ∈ 𝐵) | |
14 | 13 | con3i 150 | . . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐵 → ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) |
15 | ndmovg 6817 | . . . . . 6 ⊢ ((dom 𝐹 = (𝐴 × 𝐵) ∧ ¬ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) = ∅) | |
16 | 12, 14, 15 | syl2an 494 | . . . . 5 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) = ∅) |
17 | 10, 16 | eqtr4d 2659 | . . . 4 ⊢ (((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐷 ∈ 𝐵) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)) |
18 | 17 | ex 450 | . . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (¬ 𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷))) |
19 | oveq2 6658 | . . . 4 ⊢ (𝑥 = 𝐷 → (𝐶𝐹𝑥) = (𝐶𝐹𝐷)) | |
20 | ovex 6678 | . . . 4 ⊢ (𝐶𝐹𝐷) ∈ V | |
21 | 19, 4, 20 | fvmpt 6282 | . . 3 ⊢ (𝐷 ∈ 𝐵 → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)) |
22 | 18, 21 | pm2.61d2 172 | . 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → ((𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))‘𝐷) = (𝐶𝐹𝐷)) |
23 | 3, 22 | eqtrd 2656 | 1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴) → (𝐺‘𝐷) = (𝐶𝐹𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {csn 4177 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 dom cdm 5114 ↾ cres 5116 ∘ ccom 5118 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 2nd c2nd 7167 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: nvinvfval 27495 hhssabloilem 28118 |
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