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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcfval4 | Structured version Visualization version GIF version |
Description: The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
ofcfval4.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
ofcfval4.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcfval4.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
ofcfval4 | ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcfval4.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | fdmd 6746 | . . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
3 | 2 | mpteq1d 5242 | . 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
4 | ofcfval4.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | 1, 4 | fexd 7246 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | ofcfval4.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
7 | ofcfval3 34082 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶))) | |
8 | 5, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
9 | 1 | ffvelcdmda 7103 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵) |
10 | 1 | feqmptd 6976 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) |
11 | eqidd 2735 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶))) | |
12 | oveq1 7437 | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥𝑅𝐶) = ((𝐹‘𝑦)𝑅𝐶)) | |
13 | 9, 10, 11, 12 | fmptco 7148 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
14 | 3, 8, 13 | 3eqtr4d 2784 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ↦ cmpt 5230 dom cdm 5688 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∘f/c cofc 34075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-ofc 34076 |
This theorem is referenced by: rrvmulc 34434 |
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