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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 6719 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | off.3 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
4 | 3 | ffnd 6719 | . . 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 2734 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
9 | eqidd 2734 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
10 | 2, 4, 5, 6, 7, 8, 9 | offval 7679 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
11 | inss1 4229 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
12 | 7, 11 | eqsstrri 4018 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
13 | 12 | sseli 3979 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
14 | ffvelcdm 7084 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
15 | 1, 13, 14 | syl2an 597 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
16 | inss2 4230 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
17 | 7, 16 | eqsstrri 4018 | . . . . 5 ⊢ 𝐶 ⊆ 𝐵 |
18 | 17 | sseli 3979 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
19 | ffvelcdm 7084 | . . . 4 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
20 | 3, 18, 19 | syl2an 597 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
21 | off.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
22 | 21 | ralrimivva 3201 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
23 | 22 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
24 | ovrspc2v 7435 | . . 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 837 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
26 | 10, 25 | fmpt3d 7116 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⟶wf 6540 ‘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: suppofssd 8188 o1of2 15557 ghmplusg 19714 gsumzaddlem 19789 gsumzadd 19790 lcomf 20511 frlmup1 21353 psrbagaddcl 21481 psrbagaddclOLD 21482 psraddcl 21502 psrvscacl 21512 psrbagev1 21638 psrbagev1OLD 21639 evlslem3 21643 mndvcl 21893 tsmsadd 23651 mbfmulc2lem 25164 mbfaddlem 25177 i1fadd 25212 i1fmul 25213 itg1addlem4 25216 itg1addlem4OLD 25217 i1fmulclem 25220 i1fmulc 25221 mbfi1flimlem 25240 itg2mulclem 25264 itg2mulc 25265 itg2monolem1 25268 itg2addlem 25276 dvaddbr 25455 dvmulbr 25456 dvaddf 25459 dvmulf 25460 dv11cn 25518 plyaddlem 25729 coeeulem 25738 coeaddlem 25763 plydivlem4 25809 jensenlem2 26492 jensen 26493 basellem7 26591 basellem9 26593 dchrmulcl 26752 ofrn 31864 offinsupp1 31952 ply1degltdimlem 32707 fedgmullem1 32714 sibfof 33339 signshf 33599 circlemethhgt 33655 gg-dvmulbr 35175 poimirlem23 36511 poimirlem24 36512 poimirlem25 36513 poimirlem29 36517 poimirlem30 36518 poimirlem31 36519 poimirlem32 36520 itg2addnc 36542 ftc1anclem3 36563 ftc1anclem6 36566 ftc1anclem8 36568 lfladdcl 37941 lflvscl 37947 fsuppssind 41165 mhphf 41169 mzpclall 41465 mzpindd 41484 expgrowth 43094 binomcxplemnotnn0 43115 dvdivcncf 44643 ofaddmndmap 47019 amgmwlem 47849 |
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