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Theorem offn 7413
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 7184 . . 3 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2 eqid 2825 . . 3 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
31, 2fnmpti 6487 . 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 2826 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
10 eqidd 2826 . . . 4 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
114, 5, 6, 7, 8, 9, 10offval 7409 . . 3 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1211fneq1d 6442 . 2 (𝜑 → ((𝐹f 𝑅𝐺) Fn 𝑆 ↔ (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn 𝑆))
133, 12mpbiri 259 1 (𝜑 → (𝐹f 𝑅𝐺) Fn 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cin 3938  cmpt 5142   Fn wfn 6346  cfv 6351  (class class class)co 7151  f cof 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402
This theorem is referenced by:  offveq  7423  suppofss1d  7862  suppofss2d  7863  ofsubeq0  11627  ofnegsub  11628  ofsubge0  11629  seqof  13420  ofccat  14322  lcomfsupp  19596  psrbagcon  20072  psrbagev1  20210  frlmsslsp  20858  frlmup1  20860  i1faddlem  24211  i1fmullem  24212  dv11cn  24515  coemulc  24762  ofmulrt  24788  plydivlem3  24801  plyrem  24811  jensen  25482  basellem9  25582  broucube  34795  caofcan  40522  ofmul12  40524  ofdivrec  40525  ofdivcan4  40526  ofdivdiv2  40527  mndpsuppss  44253  mndpfsupp  44258
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