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Mirrors > Home > ILE Home > Th. List > ofres | 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 2178 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2178 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 6083 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | inss1 3355 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
10 | 5, 9 | eqsstrri 3188 | . . . 4 ⊢ 𝐶 ⊆ 𝐴 |
11 | fnssres 5324 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶) | |
12 | 1, 10, 11 | sylancl 413 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶) Fn 𝐶) |
13 | inss2 3356 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
14 | 5, 13 | eqsstrri 3188 | . . . 4 ⊢ 𝐶 ⊆ 𝐵 |
15 | fnssres 5324 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐺 ↾ 𝐶) Fn 𝐶) | |
16 | 2, 14, 15 | sylancl 413 | . . 3 ⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
17 | ssexg 4139 | . . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐶 ∈ V) | |
18 | 10, 3, 17 | sylancr 414 | . . 3 ⊢ (𝜑 → 𝐶 ∈ V) |
19 | inidm 3344 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
20 | fvres 5534 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) | |
21 | 20 | adantl 277 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥)) |
22 | fvres 5534 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) | |
23 | 22 | adantl 277 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐺 ↾ 𝐶)‘𝑥) = (𝐺‘𝑥)) |
24 | 12, 16, 18, 18, 19, 21, 23 | offval 6083 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶)) = (𝑥 ∈ 𝐶 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
25 | 8, 24 | eqtr4d 2213 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = ((𝐹 ↾ 𝐶) ∘𝑓 𝑅(𝐺 ↾ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 ↦ cmpt 4061 ↾ cres 4624 Fn wfn 5206 ‘cfv 5211 (class class class)co 5868 ∘𝑓 cof 6074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-of 6076 |
This theorem is referenced by: (None) |
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