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Mirrors > Home > MPE Home > Th. List > off | Structured version Visualization version GIF version |
Description: The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
off.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) |
off.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
off.3 | ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) |
off.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
off.5 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
off.6 | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Ref | Expression |
---|---|
off | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | off.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffnd 6601 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | off.3 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
4 | 3 | ffnd 6601 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
5 | off.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | off.5 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | off.6 | . . 3 ⊢ (𝐴 ∩ 𝐵) = 𝐶 | |
8 | eqidd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
9 | eqidd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
10 | 2, 4, 5, 6, 7, 8, 9 | offval 7542 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
11 | inss1 4162 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
12 | 7, 11 | eqsstrri 3956 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
13 | 12 | sseli 3917 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
14 | ffvelrn 6959 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
15 | 1, 13, 14 | syl2an 596 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
16 | inss2 4163 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
17 | 7, 16 | eqsstrri 3956 | . . . . 5 ⊢ 𝐶 ⊆ 𝐵 |
18 | 17 | sseli 3917 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
19 | ffvelrn 6959 | . . . 4 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
20 | 3, 18, 19 | syl2an 596 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
21 | off.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
22 | 21 | ralrimivva 3123 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
23 | 22 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
24 | ovrspc2v 7301 | . . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) | |
25 | 15, 20, 23, 24 | syl21anc 835 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
26 | 10, 25 | fmpt3d 6990 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∩ cin 3886 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∘f cof 7531 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 |
This theorem is referenced by: suppofssd 8019 o1of2 15322 ghmplusg 19447 gsumzaddlem 19522 gsumzadd 19523 lcomf 20162 frlmup1 21005 psrbagaddcl 21131 psrbagaddclOLD 21132 psraddcl 21152 psrvscacl 21162 psrbagev1 21285 psrbagev1OLD 21286 evlslem3 21290 mndvcl 21540 tsmsadd 23298 mbfmulc2lem 24811 mbfaddlem 24824 i1fadd 24859 i1fmul 24860 itg1addlem4 24863 itg1addlem4OLD 24864 i1fmulclem 24867 i1fmulc 24868 mbfi1flimlem 24887 itg2mulclem 24911 itg2mulc 24912 itg2monolem1 24915 itg2addlem 24923 dvaddbr 25102 dvmulbr 25103 dvaddf 25106 dvmulf 25107 dv11cn 25165 plyaddlem 25376 coeeulem 25385 coeaddlem 25410 plydivlem4 25456 jensenlem2 26137 jensen 26138 basellem7 26236 basellem9 26238 dchrmulcl 26397 ofrn 30976 offinsupp1 31062 fedgmullem1 31710 sibfof 32307 signshf 32567 circlemethhgt 32623 poimirlem23 35800 poimirlem24 35801 poimirlem25 35802 poimirlem29 35806 poimirlem30 35807 poimirlem31 35808 poimirlem32 35809 itg2addnc 35831 ftc1anclem3 35852 ftc1anclem6 35855 ftc1anclem8 35857 lfladdcl 37085 lflvscl 37091 fsuppssind 40282 mhphf 40285 mzpclall 40549 mzpindd 40568 expgrowth 41953 binomcxplemnotnn0 41974 dvdivcncf 43468 ofaddmndmap 45679 amgmwlem 46506 |
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