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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 6850 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) | |
4 | 2, 3 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹) |
5 | offvalfv.g | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
6 | fnfvelrn 6850 | . . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) | |
7 | 5, 6 | sylan 582 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺) |
8 | dffn5 6726 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
9 | 2, 8 | sylib 220 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
10 | dffn5 6726 | . . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) | |
11 | 5, 10 | sylib 220 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥))) |
12 | 1, 4, 7, 9, 11 | offval2 7428 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ran crn 5558 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 |
This theorem is referenced by: zlmodzxzscm 44412 zlmodzxzadd 44413 mndpsuppss 44426 lincsum 44491 |
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