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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 7168 | . . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V | |
2 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
3 | 1, 2 | fnmpti 6463 | . 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 2799 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
10 | eqidd 2799 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 7396 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
12 | 11 | fneq1d 6416 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆)) |
13 | 3, 12 | mpbiri 261 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ↦ cmpt 5110 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 |
This theorem is referenced by: offveq 7410 suppofss1d 7851 suppofss2d 7852 ofsubeq0 11622 ofnegsub 11623 ofsubge0 11624 seqof 13423 ofccat 14320 lcomfsupp 19667 frlmsslsp 20485 frlmup1 20487 psrbagcon 20609 psrbagev1 20749 i1faddlem 24297 i1fmullem 24298 dv11cn 24604 coemulc 24852 ofmulrt 24878 plydivlem3 24891 plyrem 24901 jensen 25574 basellem9 25674 broucube 35091 ofun 39416 fsuppind 39456 caofcan 41027 ofmul12 41029 ofdivrec 41030 ofdivcan4 41031 ofdivdiv2 41032 mndpsuppss 44773 mndpfsupp 44778 |
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