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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofcof | Structured version Visualization version GIF version |
Description: Relate function operation with operation with a constant. (Contributed by Thierry Arnoux, 3-Oct-2018.) |
Ref | Expression |
---|---|
ofcof.1 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
ofcof.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ofcof.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
ofcof | ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofcof.1 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | ffnd 6738 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | ofcof.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | ofcof.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | eqidd 2736 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
6 | 2, 3, 4, 5 | ofcfval 34079 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
7 | fnconstg 6797 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) |
9 | inidm 4235 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
10 | fvconst2g 7222 | . . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) | |
11 | 4, 10 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) |
12 | 2, 8, 3, 3, 9, 5, 11 | offval 7706 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
13 | 6, 12 | eqtr4d 2778 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 ↦ cmpt 5231 × cxp 5687 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ∘f/c cofc 34076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofc 34077 |
This theorem is referenced by: ofcccat 34537 |
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