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Theorem bdcriota 12771
Description: A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
Hypotheses
Ref Expression
bdcriota.bd  |- BOUNDED  ph
bdcriota.ex  |-  E! x  e.  y  ph
Assertion
Ref Expression
bdcriota  |- BOUNDED  ( iota_ x  e.  y 
ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bdcriota
Dummy variables  z  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdcriota.bd . . . . . . . . 9  |- BOUNDED  ph
21ax-bdsb 12710 . . . . . . . 8  |- BOUNDED  [ z  /  x ] ph
3 ax-bdel 12709 . . . . . . . 8  |- BOUNDED  t  e.  z
42, 3ax-bdim 12702 . . . . . . 7  |- BOUNDED  ( [ z  /  x ] ph  ->  t  e.  z )
54ax-bdal 12706 . . . . . 6  |- BOUNDED  A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )
6 df-ral 2395 . . . . . . . . 9  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
7 impexp 261 . . . . . . . . . . 11  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
87bicomi 131 . . . . . . . . . 10  |-  ( ( z  e.  y  -> 
( [ z  /  x ] ph  ->  t  e.  z ) )  <->  ( (
z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
98albii 1429 . . . . . . . . 9  |-  ( A. z ( z  e.  y  ->  ( [
z  /  x ] ph  ->  t  e.  z ) )  <->  A. z
( ( z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
106, 9bitri 183 . . . . . . . 8  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( ( z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
11 sban 1904 . . . . . . . . . . . 12  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( [
z  /  x ]
x  e.  y  /\  [ z  /  x ] ph ) )
12 clelsb3 2219 . . . . . . . . . . . . 13  |-  ( [ z  /  x ]
x  e.  y  <->  z  e.  y )
1312anbi1i 451 . . . . . . . . . . . 12  |-  ( ( [ z  /  x ] x  e.  y  /\  [ z  /  x ] ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1411, 13bitri 183 . . . . . . . . . . 11  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1514bicomi 131 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  [ z  /  x ] ph )  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
1615imbi1i 237 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
1716albii 1429 . . . . . . . 8  |-  ( A. z ( ( z  e.  y  /\  [
z  /  x ] ph )  ->  t  e.  z )  <->  A. z
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
1810, 17bitri 183 . . . . . . 7  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
19 df-clab 2102 . . . . . . . . . 10  |-  ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
2019bicomi 131 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  z  e.  { x  |  ( x  e.  y  /\  ph ) } )
2120imbi1i 237 . . . . . . . 8  |-  ( ( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z )  <-> 
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
2221albii 1429 . . . . . . 7  |-  ( A. z ( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z )  <->  A. z
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
2318, 22bitri 183 . . . . . 6  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
245, 23bd0 12712 . . . . 5  |- BOUNDED  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z )
2524bdcab 12737 . . . 4  |- BOUNDED  { t  |  A. z ( z  e. 
{ x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
26 df-int 3738 . . . 4  |-  |^| { x  |  ( x  e.  y  /\  ph ) }  =  { t  |  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
2725, 26bdceqir 12732 . . 3  |- BOUNDED 
|^| { x  |  ( x  e.  y  /\  ph ) }
28 bdcriota.ex . . . . 5  |-  E! x  e.  y  ph
29 df-reu 2397 . . . . 5  |-  ( E! x  e.  y  ph  <->  E! x ( x  e.  y  /\  ph )
)
3028, 29mpbi 144 . . . 4  |-  E! x
( x  e.  y  /\  ph )
31 iotaint 5059 . . . 4  |-  ( E! x ( x  e.  y  /\  ph )  ->  ( iota x ( x  e.  y  /\  ph ) )  =  |^| { x  |  ( x  e.  y  /\  ph ) } )
3230, 31ax-mp 7 . . 3  |-  ( iota
x ( x  e.  y  /\  ph )
)  =  |^| { x  |  ( x  e.  y  /\  ph ) }
3327, 32bdceqir 12732 . 2  |- BOUNDED  ( iota x ( x  e.  y  /\  ph ) )
34 df-riota 5684 . 2  |-  ( iota_ x  e.  y  ph )  =  ( iota x
( x  e.  y  /\  ph ) )
3533, 34bdceqir 12732 1  |- BOUNDED  ( iota_ x  e.  y 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314    e. wcel 1463   [wsb 1718   E!weu 1975   {cab 2101   A.wral 2390   E!wreu 2392   |^|cint 3737   iotacio 5044   iota_crio 5683  BOUNDED wbd 12700  BOUNDED wbdc 12728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12701  ax-bdim 12702  ax-bdal 12706  ax-bdel 12709  ax-bdsb 12710
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-sn 3499  df-pr 3500  df-uni 3703  df-int 3738  df-iota 5046  df-riota 5684  df-bdc 12729
This theorem is referenced by: (None)
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