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Theorem bdcriota 10390
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 10329 . . . . . . . 8  |- BOUNDED  [ z  /  x ] ph
3 ax-bdel 10328 . . . . . . . 8  |- BOUNDED  t  e.  z
42, 3ax-bdim 10321 . . . . . . 7  |- BOUNDED  ( [ z  /  x ] ph  ->  t  e.  z )
54ax-bdal 10325 . . . . . 6  |- BOUNDED  A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )
6 df-ral 2328 . . . . . . . . 9  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
7 impexp 254 . . . . . . . . . . 11  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
87bicomi 127 . . . . . . . . . 10  |-  ( ( z  e.  y  -> 
( [ z  /  x ] ph  ->  t  e.  z ) )  <->  ( (
z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
98albii 1375 . . . . . . . . 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 177 . . . . . . . 8  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( ( z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
11 sban 1845 . . . . . . . . . . . 12  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( [
z  /  x ]
x  e.  y  /\  [ z  /  x ] ph ) )
12 clelsb3 2158 . . . . . . . . . . . . 13  |-  ( [ z  /  x ]
x  e.  y  <->  z  e.  y )
1312anbi1i 439 . . . . . . . . . . . 12  |-  ( ( [ z  /  x ] x  e.  y  /\  [ z  /  x ] ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1411, 13bitri 177 . . . . . . . . . . 11  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1514bicomi 127 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  [ z  /  x ] ph )  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
1615imbi1i 231 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
1716albii 1375 . . . . . . . 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 177 . . . . . . 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 2043 . . . . . . . . . 10  |-  ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
2019bicomi 127 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  z  e.  { x  |  ( x  e.  y  /\  ph ) } )
2120imbi1i 231 . . . . . . . 8  |-  ( ( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z )  <-> 
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
2221albii 1375 . . . . . . 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 177 . . . . . 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 10331 . . . . 5  |- BOUNDED  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z )
2524bdcab 10356 . . . 4  |- BOUNDED  { t  |  A. z ( z  e. 
{ x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
26 df-int 3644 . . . 4  |-  |^| { x  |  ( x  e.  y  /\  ph ) }  =  { t  |  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
2725, 26bdceqir 10351 . . 3  |- BOUNDED 
|^| { x  |  ( x  e.  y  /\  ph ) }
28 bdcriota.ex . . . . 5  |-  E! x  e.  y  ph
29 df-reu 2330 . . . . 5  |-  ( E! x  e.  y  ph  <->  E! x ( x  e.  y  /\  ph )
)
3028, 29mpbi 137 . . . 4  |-  E! x
( x  e.  y  /\  ph )
31 iotaint 4908 . . . 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 10351 . 2  |- BOUNDED  ( iota x ( x  e.  y  /\  ph ) )
34 df-riota 5496 . 2  |-  ( iota_ x  e.  y  ph )  =  ( iota x
( x  e.  y  /\  ph ) )
3533, 34bdceqir 10351 1  |- BOUNDED  ( iota_ x  e.  y 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259    e. wcel 1409   [wsb 1661   E!weu 1916   {cab 2042   A.wral 2323   E!wreu 2325   |^|cint 3643   iotacio 4893   iota_crio 5495  BOUNDED wbd 10319  BOUNDED wbdc 10347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bd0 10320  ax-bdim 10321  ax-bdal 10325  ax-bdel 10328  ax-bdsb 10329
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-iota 4895  df-riota 5496  df-bdc 10348
This theorem is referenced by: (None)
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