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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 2725 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2725 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7673 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | inss1 4221 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
10 | 5, 9 | eqsstrri 4010 | . . . 4 ⊢ 𝐶 ⊆ 𝐴 |
11 | fnssres 6664 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶) | |
12 | 1, 10, 11 | sylancl 585 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶) |
13 | inss2 4222 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
14 | 5, 13 | eqsstrri 4010 | . . . 4 ⊢ 𝐶 ⊆ 𝐵 |
15 | fnssres 6664 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶) | |
16 | 2, 14, 15 | sylancl 585 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
17 | ssexg 5314 | . . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) | |
18 | 10, 3, 17 | sylancr 586 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
19 | inidm 4211 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
20 | fvres 6901 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
21 | 20 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) |
22 | fvres 6901 | . . . 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 7673 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
25 | 8, 24 | eqtr4d 2767 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘f 𝑅(𝐺 ↾ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∩ cin 3940 ⊆ wss 3941 ↦ cmpt 5222 ↾ cres 5669 Fn wfn 6529 ‘cfv 6534 (class class class)co 7402 ∘f cof 7662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 |
This theorem is referenced by: ofoafg 42653 |
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