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Mathbox for Alexander van der Vekens |
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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 6580 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) | |
4 | 2, 3 | sylan 576 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) |
5 | offvalfv.g | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
6 | fnfvelrn 6580 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) | |
7 | 5, 6 | sylan 576 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) |
8 | dffn5 6464 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
9 | 2, 8 | sylib 210 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
10 | dffn5 6464 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
11 | 5, 10 | sylib 210 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
12 | 1, 4, 7, 9, 11 | offval2 7146 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ↦ cmpt 4920 ran crn 5311 Fn wfn 6094 ‘cfv 6099 (class class class)co 6876 ∘𝑓 cof 7127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 |
This theorem is referenced by: zlmodzxzscm 42922 zlmodzxzadd 42923 mndpsuppss 42939 lincsum 43005 |
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