![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ofres | Structured version Visualization version GIF version |
Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
ofres.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
ofres.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
ofres.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofres.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
ofres.5 | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Ref | Expression |
---|---|
ofres | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofres.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | ofres.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | ofres.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | ofres.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | ofres.5 | . . 3 ⊢ (𝐴 ∩ 𝐵) = 𝐶 | |
6 | eqidd 2729 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2729 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7694 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | inss1 4229 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
10 | 5, 9 | eqsstrri 4015 | . . . 4 ⊢ 𝐶 ⊆ 𝐴 |
11 | fnssres 6678 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶) | |
12 | 1, 10, 11 | sylancl 585 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶) |
13 | inss2 4230 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
14 | 5, 13 | eqsstrri 4015 | . . . 4 ⊢ 𝐶 ⊆ 𝐵 |
15 | fnssres 6678 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶) | |
16 | 2, 14, 15 | sylancl 585 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
17 | ssexg 5323 | . . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) | |
18 | 10, 3, 17 | sylancr 586 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
19 | inidm 4219 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
20 | fvres 6916 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
21 | 20 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) |
22 | fvres 6916 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) | |
23 | 22 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) |
24 | 12, 16, 18, 18, 19, 21, 23 | offval 7694 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
25 | 8, 24 | eqtr4d 2771 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ↦ cmpt 5231 ↾ cres 5680 Fn wfn 6543 ‘cfv 6548 (class class class)co 7420 ∘f cof 7683 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 |
This theorem is referenced by: ofoafg 42783 |
Copyright terms: Public domain | W3C validator |