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Theorem riotaclbBAD 32618
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 riotacl 6079 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
3 undefnel2 6808 . . . . 5  |-  ( A  e.  _V  ->  -.  ( Undef `  A )  e.  A )
4 iffalse 3811 . . . . . . . 8  |-  ( -.  E! x  e.  A  ph 
->  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  (
Undef `  { x  |  x  e.  A }
) )
5 ax-riotaBAD 32616 . . . . . . . 8  |-  ( iota_ x  e.  A  ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
6 abid2 2567 . . . . . . . . . 10  |-  { x  |  x  e.  A }  =  A
76eqcomi 2447 . . . . . . . . 9  |-  A  =  { x  |  x  e.  A }
87fveq2i 5706 . . . . . . . 8  |-  ( Undef `  A )  =  (
Undef `  { x  |  x  e.  A }
)
94, 5, 83eqtr4g 2500 . . . . . . 7  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  ( Undef `  A ) )
109eleq1d 2509 . . . . . 6  |-  ( -.  E! x  e.  A  ph 
->  ( ( iota_ x  e.  A  ph )  e.  A  <->  ( Undef `  A
)  e.  A ) )
1110notbid 294 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( -.  ( iota_ x  e.  A  ph )  e.  A  <->  -.  ( Undef `  A )  e.  A
) )
123, 11syl5ibrcom 222 . . . 4  |-  ( A  e.  _V  ->  ( -.  E! x  e.  A  ph 
->  -.  ( iota_ x  e.  A  ph )  e.  A ) )
1312con4d 105 . . 3  |-  ( A  e.  _V  ->  (
( iota_ x  e.  A  ph )  e.  A  ->  E! x  e.  A  ph ) )
142, 13impbid2 204 . 2  |-  ( A  e.  _V  ->  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A  ph )  e.  A ) )
151, 14ax-mp 5 1  |-  ( E! x  e.  A  ph  <->  (
iota_ x  e.  A  ph )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    e. wcel 1756   {cab 2429   E!wreu 2729   _Vcvv 2984   ifcif 3803   iotacio 5391   ` cfv 5430   iota_crio 6063   Undefcund 6803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-riotaBAD 32616
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-riota 6064  df-undef 6804
This theorem is referenced by:  glbconN  33033  cdlemk36  34569
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