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 6882 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
3 | ofresid.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
4 | 3 | ffvelrnda 6882 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
5 | 2, 4 | opelxpd 5574 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵)) |
6 | 5 | fvresd 6715 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
7 | 6 | eqcomd 2742 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
8 | df-ov 7194 | . . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
9 | df-ov 7194 | . . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
10 | 7, 8, 9 | 3eqtr4g 2796 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))) |
11 | 10 | mpteq2dva 5135 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
12 | 1 | ffnd 6524 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
13 | 3 | ffnd 6524 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) |
14 | ofresid.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
15 | inidm 4119 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
16 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
17 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
18 | 12, 13, 14, 14, 15, 16, 17 | offval 7455 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
19 | 12, 13, 14, 14, 15, 16, 17 | offval 7455 | . 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
20 | 11, 18, 19 | 3eqtr4d 2781 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 ↦ cmpt 5120 × cxp 5534 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 |
This theorem is referenced by: sitmcl 31984 |
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