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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 7089 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) | |
4 | 2, 3 | sylan 578 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) |
5 | offvalfv.g | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
6 | fnfvelrn 7089 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) | |
7 | 5, 6 | sylan 578 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) |
8 | dffn5 6956 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
9 | 2, 8 | sylib 217 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
10 | dffn5 6956 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
11 | 5, 10 | sylib 217 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
12 | 1, 4, 7, 9, 11 | offval2 7705 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5232 ran crn 5679 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 |
This theorem is referenced by: zlmodzxzscm 47607 zlmodzxzadd 47608 mndpsuppss 47621 lincsum 47683 |
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