![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > offn | Structured version Visualization version GIF version |
Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offval.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
offval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offval.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
offval.5 | ⊢ (𝐴 ∩ 𝐵) = 𝑆 |
Ref | Expression |
---|---|
offn | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7457 | . . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V | |
2 | eqid 2726 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
3 | 1, 2 | fnmpti 6704 | . 2 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆 |
4 | offval.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | offval.5 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
9 | eqidd 2727 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
10 | eqidd 2727 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 7699 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
12 | 11 | fneq1d 6653 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆)) |
13 | 3, 12 | mpbiri 257 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ↦ cmpt 5236 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 ∘f cof 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 |
This theorem is referenced by: offun 7704 offveq 7715 suppofss1d 8219 suppofss2d 8220 ofsubeq0 12261 ofnegsub 12262 ofsubge0 12263 seqof 14079 ofccat 14974 frlmsslsp 21794 frlmup1 21796 psrbagcon 21927 psrbagconOLD 21928 psdmul 22160 i1faddlem 25713 i1fmullem 25714 dv11cn 26025 coemulc 26282 ofmulrt 26309 plydivlem3 26323 plyrem 26333 jensen 27017 basellem9 27117 1arithidomlem2 33411 ply1degltdimlem 33517 broucube 37355 ofun 41960 fsuppind 42062 ofoafg 43020 ofoafo 43022 ofoaid1 43024 ofoaid2 43025 ofoaass 43026 ofoacom 43027 naddcnff 43028 naddcnffo 43030 naddcnfcom 43032 naddcnfid1 43033 naddcnfass 43035 caofcan 43997 ofmul12 43999 ofdivrec 44000 ofdivcan4 44001 ofdivdiv2 44002 |
Copyright terms: Public domain | W3C validator |