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 6850 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
3 | ofresid.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
4 | 3 | ffvelrnda 6850 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
5 | 2, 4 | opelxpd 5592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵)) |
6 | 5 | fvresd 6689 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
7 | 6 | eqcomd 2827 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
8 | df-ov 7158 | . . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
9 | df-ov 7158 | . . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
10 | 7, 8, 9 | 3eqtr4g 2881 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))) |
11 | 10 | mpteq2dva 5160 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
12 | 1 | ffnd 6514 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
13 | 3 | ffnd 6514 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) |
14 | ofresid.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
15 | inidm 4194 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
16 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
17 | eqidd 2822 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
18 | 12, 13, 14, 14, 15, 16, 17 | offval 7415 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
19 | 12, 13, 14, 14, 15, 16, 17 | offval 7415 | . 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
20 | 11, 18, 19 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4572 ↦ cmpt 5145 × cxp 5552 ↾ cres 5556 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ∘f cof 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 |
This theorem is referenced by: sitmcl 31609 |
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