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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 6910 | . . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V | |
2 | eqid 2799 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
3 | 1, 2 | fnmpti 6233 | . 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 2800 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
10 | eqidd 2800 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 7138 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
12 | 11 | fneq1d 6192 | . 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆)) |
13 | 3, 12 | mpbiri 250 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 ↦ cmpt 4922 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 ∘𝑓 cof 7129 |
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-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
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 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 |
This theorem is referenced by: offveq 7152 suppofss1d 7570 suppofss2d 7571 ofsubeq0 11309 ofnegsub 11310 ofsubge0 11311 seqof 13112 ofccat 14051 lcomfsupp 19221 psrbagcon 19694 psrbagev1 19832 frlmsslsp 20460 frlmup1 20462 i1faddlem 23801 i1fmullem 23802 dv11cn 24105 coemulc 24352 ofmulrt 24378 plydivlem3 24391 plyrem 24401 jensen 25067 basellem9 25167 broucube 33932 caofcan 39304 ofmul12 39306 ofdivrec 39307 ofdivcan4 39308 ofdivdiv2 39309 mndpsuppss 42951 mndpfsupp 42956 |
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