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

Theorem offn 6783
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 6555 . . 3 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2 eqid 2609 . . 3 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
31, 2fnmpti 5921 . 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 2610 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
10 eqidd 2610 . . . 4 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
114, 5, 6, 7, 8, 9, 10offval 6779 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1211fneq1d 5881 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) Fn 𝑆 ↔ (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn 𝑆))
133, 12mpbiri 246 1 (𝜑 → (𝐹𝑓 𝑅𝐺) Fn 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cin 3538  cmpt 4637   Fn wfn 5785  cfv 5790  (class class class)co 6527  𝑓 cof 6770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772
This theorem is referenced by:  offveq  6793  suppofss1d  7196  suppofss2d  7197  ofsubeq0  10864  ofnegsub  10865  ofsubge0  10866  seqof  12675  ofccat  13502  lcomfsupp  18672  psrbagcon  19138  psrbagev1  19277  frlmsslsp  19896  frlmup1  19898  i1faddlem  23183  i1fmullem  23184  dv11cn  23485  coemulc  23732  ofmulrt  23758  plydivlem3  23771  plyrem  23781  jensen  24432  basellem9  24532  broucube  32409  caofcan  37340  ofmul12  37342  ofdivrec  37343  ofdivcan4  37344  ofdivdiv2  37345  mndpsuppss  41941  mndpfsupp  41946
  Copyright terms: Public domain W3C validator