MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offn Structured version   Visualization version   GIF version

Theorem offn 7710
Description: The function operation produces a function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (𝜑𝐹 Fn 𝐴)
offval.2 (𝜑𝐺 Fn 𝐵)
offval.3 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
Assertion
Ref Expression
offn (𝜑 → (𝐹f 𝑅𝐺) Fn 𝑆)

Proof of Theorem offn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 7464 . . 3 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2 eqid 2737 . . 3 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
31, 2fnmpti 6711 . 2 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn 𝑆
4 offval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.3 . . . 4 (𝜑𝐴𝑉)
7 offval.4 . . . 4 (𝜑𝐵𝑊)
8 offval.5 . . . 4 (𝐴𝐵) = 𝑆
9 eqidd 2738 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
10 eqidd 2738 . . . 4 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
114, 5, 6, 7, 8, 9, 10offval 7706 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1211fneq1d 6661 . 2 (𝜑 → ((𝐹f 𝑅𝐺) Fn 𝑆 ↔ (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn 𝑆))
133, 12mpbiri 258 1 (𝜑 → (𝐹f 𝑅𝐺) Fn 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cin 3950  cmpt 5225   Fn wfn 6556  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:  offun  7711  offveq  7723  suppofss1d  8229  suppofss2d  8230  ofsubeq0  12263  ofnegsub  12264  ofsubge0  12265  seqof  14100  ofccat  15008  frlmsslsp  21816  frlmup1  21818  psrbagcon  21945  psdmul  22170  i1faddlem  25728  i1fmullem  25729  dv11cn  26040  coemulc  26294  ofmulrt  26323  plydivlem3  26337  plyrem  26347  jensen  27032  basellem9  27132  1arithidomlem2  33564  ply1degltdimlem  33673  broucube  37661  ofun  42277  fsuppind  42600  ofoafg  43367  ofoafo  43369  ofoaid1  43371  ofoaid2  43372  ofoaass  43373  ofoacom  43374  naddcnff  43375  naddcnffo  43377  naddcnfcom  43379  naddcnfid1  43380  naddcnfass  43382  caofcan  44342  ofmul12  44344  ofdivrec  44345  ofdivcan4  44346  ofdivdiv2  44347
  Copyright terms: Public domain W3C validator