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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 3833 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
4 | 2, 3 | eqsstr3i 3636 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐴 |
5 | 4 | sseli 3599 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐴) |
6 | ffvelrn 6357 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝑆 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝑆) | |
7 | 1, 5, 6 | syl2an 494 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆) |
8 | off.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇) | |
9 | inss2 3834 | . . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
10 | 2, 9 | eqsstr3i 3636 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐵 |
11 | 10 | sseli 3599 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵) |
12 | ffvelrn 6357 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝑇 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) ∈ 𝑇) | |
13 | 8, 11, 12 | syl2an 494 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇) |
14 | off.1 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈) | |
15 | 14 | ralrimivva 2971 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
16 | 15 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) |
17 | oveq1 6657 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥𝑅𝑦) = ((𝐹‘𝑧)𝑅𝑦)) | |
18 | 17 | eleq1d 2686 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈)) |
19 | oveq2 6658 | . . . . . 6 ⊢ (𝑦 = (𝐺‘𝑧) → ((𝐹‘𝑧)𝑅𝑦) = ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
20 | 19 | eleq1d 2686 | . . . . 5 ⊢ (𝑦 = (𝐺‘𝑧) → (((𝐹‘𝑧)𝑅𝑦) ∈ 𝑈 ↔ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)) |
21 | 18, 20 | rspc2va 3323 | . . . 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 1325 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈) |
23 | eqid 2622 | . . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) | |
24 | 22, 23 | fmptd 6385 | . 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈) |
25 | ffn 6045 | . . . . 5 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
26 | 1, 25 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
27 | ffn 6045 | . . . . 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 2623 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧)) | |
32 | eqidd 2623 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧)) | |
33 | 26, 28, 29, 30, 2, 31, 32 | offval 6904 | . . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))) |
34 | 33 | feq1d 6030 | . 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)) |
35 | 24, 34 | mpbird 247 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ↦ cmpt 4729 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 |
This theorem is referenced by: o1of2 14343 ghmplusg 18249 gsumzaddlem 18321 gsumzadd 18322 lcomf 18902 psrbagaddcl 19370 psraddcl 19383 psrvscacl 19393 psrbagev1 19510 evlslem3 19514 frlmup1 20137 mndvcl 20197 tsmsadd 21950 mbfmulc2lem 23414 mbfaddlem 23427 i1fadd 23462 i1fmul 23463 itg1addlem4 23466 i1fmulclem 23469 i1fmulc 23470 mbfi1flimlem 23489 itg2mulclem 23513 itg2mulc 23514 itg2monolem1 23517 itg2addlem 23525 dvaddbr 23701 dvmulbr 23702 dvaddf 23705 dvmulf 23706 dv11cn 23764 plyaddlem 23971 coeeulem 23980 coeaddlem 24005 plydivlem4 24051 jensenlem2 24714 jensen 24715 basellem7 24813 basellem9 24815 dchrmulcl 24974 ofrn 29441 sibfof 30402 signshf 30665 circlemethhgt 30721 poimirlem23 33432 poimirlem24 33433 poimirlem25 33434 poimirlem29 33438 poimirlem30 33439 poimirlem31 33440 poimirlem32 33441 itg2addnc 33464 ftc1anclem3 33487 ftc1anclem6 33490 ftc1anclem8 33492 lfladdcl 34358 lflvscl 34364 mzpclall 37290 mzpindd 37309 expgrowth 38534 binomcxplemnotnn0 38555 dvdivcncf 40142 ofaddmndmap 42122 amgmwlem 42548 |
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