![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6717 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | off.3 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
4 | 3 | ffnd 6717 | . . 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 2728 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
9 | eqidd 2728 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
10 | 2, 4, 5, 6, 7, 8, 9 | offval 7688 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
11 | inss1 4224 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
12 | 7, 11 | eqsstrri 4013 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
13 | 12 | sseli 3974 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
14 | ffvelcdm 7085 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
15 | 1, 13, 14 | syl2an 595 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
16 | inss2 4225 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
17 | 7, 16 | eqsstrri 4013 | . . . . 5 ⊢ 𝐶 ⊆ 𝐵 |
18 | 17 | sseli 3974 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
19 | ffvelcdm 7085 | . . . 4 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
20 | 3, 18, 19 | syl2an 595 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
21 | off.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
22 | 21 | ralrimivva 3195 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
23 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
24 | ovrspc2v 7440 | . . 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 7120 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∩ cin 3943 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘f cof 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 |
This theorem is referenced by: suppofssd 8202 o1of2 15581 ghmplusg 19792 gsumzaddlem 19867 gsumzadd 19868 lcomf 20773 frlmup1 21719 psrbagaddcl 21848 psrbagaddclOLD 21849 psraddcl 21870 psraddclOLD 21871 psrvscacl 21881 psrbagev1 22008 psrbagev1OLD 22009 evlslem3 22013 mndvcl 22280 tsmsadd 24038 mbfmulc2lem 25563 mbfaddlem 25576 i1fadd 25611 i1fmul 25612 itg1addlem4 25615 itg1addlem4OLD 25616 i1fmulclem 25619 i1fmulc 25620 mbfi1flimlem 25639 itg2mulclem 25663 itg2mulc 25664 itg2monolem1 25667 itg2addlem 25675 dvaddbr 25855 dvmulbr 25856 dvmulbrOLD 25857 dvaddf 25860 dvmulf 25861 dv11cn 25921 plyaddlem 26136 coeeulem 26145 coeaddlem 26170 plydivlem4 26218 jensenlem2 26907 jensen 26908 basellem7 27006 basellem9 27008 dchrmulcl 27169 ofrn 32408 offinsupp1 32493 ply1degltdimlem 33252 fedgmullem1 33259 sibfof 33896 signshf 34156 circlemethhgt 34211 poimirlem23 37051 poimirlem24 37052 poimirlem25 37053 poimirlem29 37057 poimirlem30 37058 poimirlem31 37059 poimirlem32 37060 itg2addnc 37082 ftc1anclem3 37103 ftc1anclem6 37106 ftc1anclem8 37108 lfladdcl 38480 lflvscl 38486 fsuppssind 41748 mhphf 41752 mzpclall 42069 mzpindd 42088 expgrowth 43695 binomcxplemnotnn0 43716 dvdivcncf 45238 ofaddmndmap 47330 amgmwlem 48158 |
Copyright terms: Public domain | W3C validator |