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Mirrors > Home > MPE Home > Th. List > Mathboxes > offvalfv | Structured version Visualization version GIF version |
Description: The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
offvalfv.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offvalfv.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offvalfv.g | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
Ref | Expression |
---|---|
offvalfv | ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offvalfv.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | offvalfv.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
3 | fnfvelrn 6396 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) | |
4 | 2, 3 | sylan 487 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) |
5 | offvalfv.g | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
6 | fnfvelrn 6396 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) | |
7 | 5, 6 | sylan 487 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) |
8 | dffn5 6280 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
9 | 2, 8 | sylib 208 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
10 | dffn5 6280 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
11 | 5, 10 | sylib 208 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
12 | 1, 4, 7, 9, 11 | offval2 6956 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ↦ cmpt 4762 ran crn 5144 Fn wfn 5921 ‘cfv 5926 (class class class)co 6690 ∘𝑓 cof 6937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 |
This theorem is referenced by: zlmodzxzscm 42460 zlmodzxzadd 42461 mndpsuppss 42477 lincsum 42543 |
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