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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 7442 | . . 3 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V | |
2 | eqid 2733 | . . 3 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
3 | 1, 2 | fnmpti 6694 | . 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 2734 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
10 | eqidd 2734 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
11 | 4, 5, 6, 7, 8, 9, 10 | offval 7679 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
12 | 11 | fneq1d 6643 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) Fn 𝑆 ↔ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn 𝑆)) |
13 | 3, 12 | mpbiri 258 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3948 ↦ cmpt 5232 Fn wfn 6539 ‘cfv 6544 (class class class)co 7409 ∘f cof 7668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 |
This theorem is referenced by: offun 7684 offveq 7694 suppofss1d 8189 suppofss2d 8190 ofsubeq0 12209 ofnegsub 12210 ofsubge0 12211 seqof 14025 ofccat 14916 frlmsslsp 21351 frlmup1 21353 psrbagcon 21483 psrbagconOLD 21484 i1faddlem 25210 i1fmullem 25211 dv11cn 25518 coemulc 25769 ofmulrt 25795 plydivlem3 25808 plyrem 25818 jensen 26493 basellem9 26593 ply1degltdimlem 32707 broucube 36522 ofun 41058 fsuppind 41162 ofoafg 42104 ofoafo 42106 ofoaid1 42108 ofoaid2 42109 ofoaass 42110 ofoacom 42111 naddcnff 42112 naddcnffo 42114 naddcnfcom 42116 naddcnfid1 42117 naddcnfass 42119 caofcan 43082 ofmul12 43084 ofdivrec 43085 ofdivcan4 43086 ofdivdiv2 43087 |
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