Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
3 | ofresid.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) | |
4 | 3 | ffvelrnda 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵) |
5 | opelxp 5070 | . . . . . . 7 ⊢ (〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵) ↔ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵)) | |
6 | 2, 4, 5 | sylanbrc 695 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵)) |
7 | fvres 6117 | . . . . . 6 ⊢ (〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (𝐵 × 𝐵) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
9 | 8 | eqcomd 2616 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉)) |
10 | df-ov 6552 | . . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
11 | df-ov 6552 | . . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘〈(𝐹‘𝑥), (𝐺‘𝑥)〉) | |
12 | 9, 10, 11 | 3eqtr4g 2669 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))) |
13 | 12 | mpteq2dva 4672 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
14 | ffn 5958 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
15 | 1, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
16 | ffn 5958 | . . . 4 ⊢ (𝐺:𝐴⟶𝐵 → 𝐺 Fn 𝐴) | |
17 | 3, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) |
18 | ofresid.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
19 | inidm 3784 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
20 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
21 | eqidd 2611 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
22 | 15, 17, 18, 18, 19, 20, 21 | offval 6802 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
23 | 15, 17, 18, 18, 19, 20, 21 | offval 6802 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))) |
24 | 13, 22, 23 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: sitmcl 29740 |
Copyright terms: Public domain | W3C validator |