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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 6674 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
3 | off.3 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
4 | 3 | ffnd 6674 | . . 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 7631 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
11 | inss1 4193 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
12 | 7, 11 | eqsstrri 3984 | . . . . 5 ⊢ 𝐶 ⊆ 𝐴 |
13 | 12 | sseli 3945 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
14 | ffvelcdm 7037 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
15 | 1, 13, 14 | syl2an 597 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
16 | inss2 4194 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
17 | 7, 16 | eqsstrri 3984 | . . . . 5 ⊢ 𝐶 ⊆ 𝐵 |
18 | 17 | sseli 3945 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
19 | ffvelcdm 7037 | . . . 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 3198 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
23 | 22 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
24 | ovrspc2v 7388 | . . 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 7069 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∩ cin 3914 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: suppofssd 8139 o1of2 15502 ghmplusg 19631 gsumzaddlem 19705 gsumzadd 19706 lcomf 20377 frlmup1 21220 psrbagaddcl 21346 psrbagaddclOLD 21347 psraddcl 21367 psrvscacl 21377 psrbagev1 21501 psrbagev1OLD 21502 evlslem3 21506 mndvcl 21756 tsmsadd 23514 mbfmulc2lem 25027 mbfaddlem 25040 i1fadd 25075 i1fmul 25076 itg1addlem4 25079 itg1addlem4OLD 25080 i1fmulclem 25083 i1fmulc 25084 mbfi1flimlem 25103 itg2mulclem 25127 itg2mulc 25128 itg2monolem1 25131 itg2addlem 25139 dvaddbr 25318 dvmulbr 25319 dvaddf 25322 dvmulf 25323 dv11cn 25381 plyaddlem 25592 coeeulem 25601 coeaddlem 25626 plydivlem4 25672 jensenlem2 26353 jensen 26354 basellem7 26452 basellem9 26454 dchrmulcl 26613 ofrn 31597 offinsupp1 31686 fedgmullem1 32364 sibfof 32980 signshf 33240 circlemethhgt 33296 poimirlem23 36130 poimirlem24 36131 poimirlem25 36132 poimirlem29 36136 poimirlem30 36137 poimirlem31 36138 poimirlem32 36139 itg2addnc 36161 ftc1anclem3 36182 ftc1anclem6 36185 ftc1anclem8 36187 lfladdcl 37562 lflvscl 37568 fsuppssind 40797 mhphf 40800 mzpclall 41079 mzpindd 41098 expgrowth 42689 binomcxplemnotnn0 42710 dvdivcncf 44242 ofaddmndmap 46493 amgmwlem 47323 |
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