Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofresid | Structured version Visualization version GIF version |
Description: Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
Ref | Expression |
---|---|
ofresid.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
ofresid.2 | ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) |
ofresid.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
ofresid | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofresid.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffvelrnda 6843 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
3 | ofresid.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
4 | 3 | ffvelrnda 6843 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
5 | 2, 4 | opelxpd 5586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵)) |
6 | 5 | fvresd 6683 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
7 | 6 | eqcomd 2824 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
8 | df-ov 7148 | . . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
9 | df-ov 7148 | . . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
10 | 7, 8, 9 | 3eqtr4g 2878 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))) |
11 | 10 | mpteq2dva 5152 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
12 | 1 | ffnd 6508 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
13 | 3 | ffnd 6508 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) |
14 | ofresid.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
15 | inidm 4192 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
16 | eqidd 2819 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
17 | eqidd 2819 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
18 | 12, 13, 14, 14, 15, 16, 17 | offval 7405 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
19 | 12, 13, 14, 14, 15, 16, 17 | offval 7405 | . 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
20 | 11, 18, 19 | 3eqtr4d 2863 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 ↦ cmpt 5137 × cxp 5546 ↾ cres 5550 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 |
This theorem is referenced by: sitmcl 31508 |
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