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Theorem ofexg 5954
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 5950 . . 3 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
21mpofun 5841 . 2 Fun ∘𝑓 𝑅
3 resfunexg 5609 . 2 ((Fun ∘𝑓 𝑅𝐴𝑉) → ( ∘𝑓 𝑅𝐴) ∈ V)
42, 3mpan 420 1 (𝐴𝑉 → ( ∘𝑓 𝑅𝐴) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  Vcvv 2660  cin 3040  cmpt 3959  dom cdm 4509  cres 4511  Fun wfun 5087  cfv 5093  (class class class)co 5742  𝑓 cof 5948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-oprab 5746  df-mpo 5747  df-of 5950
This theorem is referenced by:  ofmresex  6003
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