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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 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | off.6 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) = 𝐶 | |
3 | inss1 3795 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
4 | 2, 3 | eqsstr3i 3599 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐴 |
5 | 4 | sseli 3564 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
6 | ffvelrn 6265 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
7 | 1, 5, 6 | syl2an 493 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
8 | off.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
9 | inss2 3796 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
10 | 2, 9 | eqsstr3i 3599 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐵 |
11 | 10 | sseli 3564 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
12 | ffvelrn 6265 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
13 | 8, 11, 12 | syl2an 493 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
14 | off.1 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
15 | 14 | ralrimivva 2954 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
16 | 15 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
17 | oveq1 6556 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦)) | |
18 | 17 | eleq1d 2672 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈)) |
19 | oveq2 6557 | . . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
20 | 19 | eleq1d 2672 | . . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)) |
21 | 18, 20 | rspc2va 3294 | . . . 4 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
22 | 7, 13, 16, 21 | syl21anc 1317 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
23 | eqid 2610 | . . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
24 | 22, 23 | fmptd 6292 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈) |
25 | ffn 5958 | . . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
26 | 1, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
27 | ffn 5958 | . . . . 5 ⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵) | |
28 | 8, 27 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
29 | off.4 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
30 | off.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
31 | eqidd 2611 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
32 | eqidd 2611 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 6802 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
34 | 33 | feq1d 5943 | . 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)) |
35 | 24, 34 | mpbird 246 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∩ cin 3539 ↦ cmpt 4643 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: o1of2 14191 ghmplusg 18072 gsumzaddlem 18144 gsumzadd 18145 lcomf 18725 psrbagaddcl 19191 psraddcl 19204 psrvscacl 19214 psrbagev1 19331 evlslem3 19335 frlmup1 19956 mndvcl 20016 tsmsadd 21760 mbfmulc2lem 23220 mbfaddlem 23233 i1fadd 23268 i1fmul 23269 itg1addlem4 23272 i1fmulclem 23275 i1fmulc 23276 mbfi1flimlem 23295 itg2mulclem 23319 itg2mulc 23320 itg2monolem1 23323 itg2addlem 23331 dvaddbr 23507 dvmulbr 23508 dvaddf 23511 dvmulf 23512 dv11cn 23568 plyaddlem 23775 coeeulem 23784 coeaddlem 23809 plydivlem4 23855 jensenlem2 24514 jensen 24515 basellem7 24613 basellem9 24615 dchrmulcl 24774 ofrn 28821 sibfof 29729 signshf 29991 poimirlem23 32602 poimirlem24 32603 poimirlem25 32604 poimirlem29 32608 poimirlem30 32609 poimirlem31 32610 poimirlem32 32611 itg2addnc 32634 ftc1anclem3 32657 ftc1anclem6 32660 ftc1anclem8 32662 lfladdcl 33376 lflvscl 33382 mzpclall 36308 mzpindd 36327 expgrowth 37556 binomcxplemnotnn0 37577 dvdivcncf 38817 ofaddmndmap 41915 amgmwlem 42357 |
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