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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 6674 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | ofcof.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | ofcof.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
6 | 2, 3, 4, 5 | ofcfval 32737 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
7 | fnconstg 6735 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) |
9 | inidm 4183 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
10 | fvconst2g 7156 | . . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) | |
11 | 4, 10 | sylan 581 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) |
12 | 2, 8, 3, 3, 9, 5, 11 | offval 7631 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
13 | 6, 12 | eqtr4d 2780 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4591 ↦ cmpt 5193 × cxp 5636 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 ∘f/c cofc 32734 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofc 32735 |
This theorem is referenced by: ofcccat 33195 |
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