Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 6847 | . . 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 6847 | . . 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 6723 | . . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) | |
9 | 2, 8 | sylib 220 | . 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
10 | dffn5 6723 | . . 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 7425 | 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 2110 ↦ cmpt 5145 ran crn 5555 Fn wfn 6349 ‘cfv 6354 (class class class)co 7155 ∘f cof 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 |
This theorem is referenced by: zlmodzxzscm 44404 zlmodzxzadd 44405 mndpsuppss 44418 lincsum 44483 |
Copyright terms: Public domain | W3C validator |