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Theorem riotaclb 6591
Description: Closure of restricted iota. (Contributed by NM, 15-Sep-2011.)
Hypothesis
Ref Expression
riotaclb.1  |-  A  e. 
_V
Assertion
Ref Expression
riotaclb  |-  ( 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 riotaclb
StepHypRef Expression
1 riotaclb.1 . 2  |-  A  e. 
_V
2 riotaclbg 6590 . 2  |-  ( A  e.  _V  ->  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  e.  A
) )
31, 2ax-mp 8 1  |-  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  e.  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    e. wcel 1726   E!wreu 2708   _Vcvv 2957   iota_crio 6543
This theorem is referenced by:  lubprop  14438  glbprop  14443  joinlem  14448  meetlem  14455  isglbd  14545  glbconN  30175  cdleme25cl  31155  cdleme29cl  31175  cdlemefrs29clN  31197  cdlemk29-3  31709  cdlemk36  31711  cdlemkid5  31733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-undef 6544  df-riota 6550
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