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Theorem fngid 23836
Description: GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
fngid  |- GId  Fn  _V

Proof of Theorem fngid
Dummy variables  u  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6155 . 2  |-  ( iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) )  e.  _V
2 df-gid 23814 . 2  |- GId  =  ( g  e.  _V  |->  (
iota_ u  e.  ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
31, 2fnmpti 5637 1  |- GId  Fn  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370   A.wral 2795   _Vcvv 3068   ran crn 4939    Fn wfn 5511   iota_crio 6150  (class class class)co 6190  GIdcgi 23809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fn 5519  df-riota 6151  df-gid 23814
This theorem is referenced by: (None)
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