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 6525 | . . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
3 | 2 | mpteq1d 5157 | . 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
4 | ofcfval4.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | fex 6991 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
6 | 1, 4, 5 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
7 | ofcfval4.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
8 | ofcfval3 31363 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶))) | |
9 | 6, 7, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
10 | 1 | ffvelrnda 6853 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵) |
11 | 1 | feqmptd 6735 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦))) |
12 | eqidd 2824 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶))) | |
13 | oveq1 7165 | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥𝑅𝐶) = ((𝐹‘𝑦)𝑅𝐶)) | |
14 | 10, 11, 12, 13 | fmptco 6893 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶))) |
15 | 3, 9, 14 | 3eqtr4d 2868 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 dom cdm 5557 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘f/c cofc 31356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-ofc 31357 |
This theorem is referenced by: rrvmulc 31713 |
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