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Theorem ofexg 6799
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 6795 . . 3 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
21mpt2fun 6660 . 2 Fun ∘𝑓 𝑅
3 resfunexg 6384 . 2 ((Fun ∘𝑓 𝑅𝐴𝑉) → ( ∘𝑓 𝑅𝐴) ∈ V)
42, 3mpan 702 1 (𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  Vcvv 3173  cin 3539  cmpt 4643  dom cdm 5038  cres 5040  Fun wfun 5798  cfv 5804  (class class class)co 6549  𝑓 cof 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-of 6795
This theorem is referenced by:  ofmresex  7056  psrplusg  19202  dchrplusg  24772
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