| 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 6737 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| 3 | off.3 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
| 4 | 3 | ffnd 6737 | . . 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 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
| 9 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
| 10 | 2, 4, 5, 6, 7, 8, 9 | offval 7706 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
| 11 | inss1 4237 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 12 | 7, 11 | eqsstrri 4031 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
| 13 | 12 | sseli 3979 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
| 14 | ffvelcdm 7101 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
| 15 | 1, 13, 14 | syl2an 596 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
| 16 | inss2 4238 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
| 17 | 7, 16 | eqsstrri 4031 | . . . . 5 ⊢ 𝐶 ⊆ 𝐵 |
| 18 | 17 | sseli 3979 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
| 19 | ffvelcdm 7101 | . . . 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 3202 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
| 23 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
| 24 | ovrspc2v 7457 | . . 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 838 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
| 26 | 10, 25 | fmpt3d 7136 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
| This theorem is referenced by: suppofssd 8228 o1of2 15649 mndvcl 18810 ghmplusg 19864 gsumzaddlem 19939 gsumzadd 19940 lcomf 20899 frlmup1 21818 psrbagaddcl 21944 psraddcl 21958 psraddclOLD 21959 psrvscacl 21971 psrbagev1 22101 evlslem3 22104 tsmsadd 24155 mbfmulc2lem 25682 mbfaddlem 25695 i1fadd 25730 i1fmul 25731 itg1addlem4 25734 i1fmulclem 25737 i1fmulc 25738 mbfi1flimlem 25757 itg2mulclem 25781 itg2mulc 25782 itg2monolem1 25785 itg2addlem 25793 dvaddbr 25974 dvmulbr 25975 dvmulbrOLD 25976 dvaddf 25979 dvmulf 25980 dv11cn 26040 plyaddlem 26254 coeeulem 26263 coeaddlem 26288 plydivlem4 26338 jensenlem2 27031 jensen 27032 basellem7 27130 basellem9 27132 dchrmulcl 27293 ofrn 32649 offinsupp1 32738 elrgspnlem1 33246 1arithidomlem2 33564 1arithidom 33565 ply1degltdimlem 33673 fedgmullem1 33680 sibfof 34342 signshf 34603 circlemethhgt 34658 poimirlem23 37650 poimirlem24 37651 poimirlem25 37652 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 poimirlem32 37659 itg2addnc 37681 ftc1anclem3 37702 ftc1anclem6 37705 ftc1anclem8 37707 lfladdcl 39072 lflvscl 39078 fsuppssind 42603 mhphf 42607 mzpclall 42738 mzpindd 42757 expgrowth 44354 binomcxplemnotnn0 44375 dvdivcncf 45942 ofaddmndmap 48259 amgmwlem 49321 |
| Copyright terms: Public domain | W3C validator |