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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 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) | 
| 3 | ofresid.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
| 4 | 3 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) | 
| 5 | 2, 4 | opelxpd 5724 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵)) | 
| 6 | 5 | fvresd 6926 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) | 
| 7 | 6 | eqcomd 2743 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) | 
| 8 | df-ov 7434 | . . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
| 9 | df-ov 7434 | . . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
| 10 | 7, 8, 9 | 3eqtr4g 2802 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))) | 
| 11 | 10 | mpteq2dva 5242 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) | 
| 12 | 1 | ffnd 6737 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | 
| 13 | 3 | ffnd 6737 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | 
| 14 | ofresid.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 15 | inidm 4227 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 16 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
| 17 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
| 18 | 12, 13, 14, 14, 15, 16, 17 | offval 7706 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) | 
| 19 | 12, 13, 14, 14, 15, 16, 17 | offval 7706 | . 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) | 
| 20 | 11, 18, 19 | 3eqtr4d 2787 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 | 
| This theorem is referenced by: sitmcl 34353 | 
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