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Theorem bdcriota 11418
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 11357 . . . . . . . 8  |- BOUNDED  [ z  /  x ] ph
3 ax-bdel 11356 . . . . . . . 8  |- BOUNDED  t  e.  z
42, 3ax-bdim 11349 . . . . . . 7  |- BOUNDED  ( [ z  /  x ] ph  ->  t  e.  z )
54ax-bdal 11353 . . . . . 6  |- BOUNDED  A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )
6 df-ral 2364 . . . . . . . . 9  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
7 impexp 259 . . . . . . . . . . 11  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
87bicomi 130 . . . . . . . . . 10  |-  ( ( z  e.  y  -> 
( [ z  /  x ] ph  ->  t  e.  z ) )  <->  ( (
z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
98albii 1404 . . . . . . . . 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 182 . . . . . . . 8  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( ( z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
11 sban 1877 . . . . . . . . . . . 12  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( [
z  /  x ]
x  e.  y  /\  [ z  /  x ] ph ) )
12 clelsb3 2192 . . . . . . . . . . . . 13  |-  ( [ z  /  x ]
x  e.  y  <->  z  e.  y )
1312anbi1i 446 . . . . . . . . . . . 12  |-  ( ( [ z  /  x ] x  e.  y  /\  [ z  /  x ] ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1411, 13bitri 182 . . . . . . . . . . 11  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1514bicomi 130 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  [ z  /  x ] ph )  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
1615imbi1i 236 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
1716albii 1404 . . . . . . . 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 182 . . . . . . 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 2075 . . . . . . . . . 10  |-  ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
2019bicomi 130 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  z  e.  { x  |  ( x  e.  y  /\  ph ) } )
2120imbi1i 236 . . . . . . . 8  |-  ( ( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z )  <-> 
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
2221albii 1404 . . . . . . 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 182 . . . . . 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 11359 . . . . 5  |- BOUNDED  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z )
2524bdcab 11384 . . . 4  |- BOUNDED  { t  |  A. z ( z  e. 
{ x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
26 df-int 3684 . . . 4  |-  |^| { x  |  ( x  e.  y  /\  ph ) }  =  { t  |  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
2725, 26bdceqir 11379 . . 3  |- BOUNDED 
|^| { x  |  ( x  e.  y  /\  ph ) }
28 bdcriota.ex . . . . 5  |-  E! x  e.  y  ph
29 df-reu 2366 . . . . 5  |-  ( E! x  e.  y  ph  <->  E! x ( x  e.  y  /\  ph )
)
3028, 29mpbi 143 . . . 4  |-  E! x
( x  e.  y  /\  ph )
31 iotaint 4980 . . . 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 11379 . 2  |- BOUNDED  ( iota x ( x  e.  y  /\  ph ) )
34 df-riota 5590 . 2  |-  ( iota_ x  e.  y  ph )  =  ( iota x
( x  e.  y  /\  ph ) )
3533, 34bdceqir 11379 1  |- BOUNDED  ( iota_ x  e.  y 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   [wsb 1692   E!weu 1948   {cab 2074   A.wral 2359   E!wreu 2361   |^|cint 3683   iotacio 4965   iota_crio 5589  BOUNDED wbd 11347  BOUNDED wbdc 11375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11348  ax-bdim 11349  ax-bdal 11353  ax-bdel 11356  ax-bdsb 11357
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-iota 4967  df-riota 5590  df-bdc 11376
This theorem is referenced by: (None)
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