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Theorem riotaclbBAD 34787
Description: Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaclb.1  |-  A  e. 
_V
Assertion
Ref Expression
riotaclbBAD  |-  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A  ph )  e.  A )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotaclbBAD
StepHypRef Expression
1 riotaclb.1 . 2  |-  A  e. 
_V
2 riotaclbgBAD 34786 . 2  |-  ( A  e.  _V  ->  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A  ph )  e.  A ) )
31, 2ax-mp 5 1  |-  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A  ph )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1819   E!wreu 2809   _Vcvv 3109   iota_crio 6257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-riota 6258  df-undef 7020
This theorem is referenced by:  glbconN  35202  cdlemk36  36740
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