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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 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7618 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | inss1 4186 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
10 | 5, 9 | eqsstrri 3977 | . . . 4 ⊢ 𝐶 ⊆ 𝐴 |
11 | fnssres 6621 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶) | |
12 | 1, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶) |
13 | inss2 4187 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
14 | 5, 13 | eqsstrri 3977 | . . . 4 ⊢ 𝐶 ⊆ 𝐵 |
15 | fnssres 6621 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶) | |
16 | 2, 14, 15 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
17 | ssexg 5278 | . . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) | |
18 | 10, 3, 17 | sylancr 587 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
19 | inidm 4176 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
20 | fvres 6858 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
21 | 20 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) |
22 | fvres 6858 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) | |
23 | 22 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) |
24 | 12, 16, 18, 18, 19, 21, 23 | offval 7618 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
25 | 8, 24 | eqtr4d 2780 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∩ cin 3907 ⊆ wss 3908 ↦ cmpt 5186 ↾ cres 5633 Fn wfn 6488 ‘cfv 6493 (class class class)co 7351 ∘f cof 7607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 |
This theorem is referenced by: ofoafg 41568 |
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