Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 6524 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | ofcof.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | ofcof.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
5 | eqidd 2737 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
6 | 2, 3, 4, 5 | ofcfval 31732 | . 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
7 | fnconstg 6585 | . . . 4 ⊢ (𝐶 ∈ 𝑊 → (𝐴 × {𝐶}) Fn 𝐴) | |
8 | 4, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 × {𝐶}) Fn 𝐴) |
9 | inidm 4119 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
10 | fvconst2g 6995 | . . . 4 ⊢ ((𝐶 ∈ 𝑊 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) | |
11 | 4, 10 | sylan 583 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {𝐶})‘𝑥) = 𝐶) |
12 | 2, 8, 3, 3, 9, 5, 11 | offval 7455 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐶})) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶))) |
13 | 6, 12 | eqtr4d 2774 | 1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f 𝑅(𝐴 × {𝐶}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {csn 4527 ↦ cmpt 5120 × cxp 5534 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ∘f cof 7445 ∘f/c cofc 31729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofc 31730 |
This theorem is referenced by: ofcccat 32188 |
Copyright terms: Public domain | W3C validator |