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Theorem ofexg 7690
Description: A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
Assertion
Ref Expression
ofexg (𝐴𝑉 → ( ∘f 𝑅𝐴) ∈ V)

Proof of Theorem ofexg
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-of 7685 . . 3 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
21mpofun 7544 . 2 Fun ∘f 𝑅
3 resfunexg 7227 . 2 ((Fun ∘f 𝑅𝐴𝑉) → ( ∘f 𝑅𝐴) ∈ V)
42, 3mpan 689 1 (𝐴𝑉 → ( ∘f 𝑅𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471  cin 3946  cmpt 5231  dom cdm 5678  cres 5680  Fun wfun 6542  cfv 6548  (class class class)co 7420  f cof 7683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-oprab 7424  df-mpo 7425  df-of 7685
This theorem is referenced by:  ofmresex  7989  psrplusg  21880  dchrplusg  27179
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