Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcriota Unicode version

Theorem bdcriota 14720
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 14659 . . . . . . . 8  |- BOUNDED  [ z  /  x ] ph
3 ax-bdel 14658 . . . . . . . 8  |- BOUNDED  t  e.  z
42, 3ax-bdim 14651 . . . . . . 7  |- BOUNDED  ( [ z  /  x ] ph  ->  t  e.  z )
54ax-bdal 14655 . . . . . 6  |- BOUNDED  A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )
6 df-ral 2460 . . . . . . . . 9  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
7 impexp 263 . . . . . . . . . . 11  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( z  e.  y  ->  ( [ z  /  x ] ph  ->  t  e.  z ) ) )
87bicomi 132 . . . . . . . . . 10  |-  ( ( z  e.  y  -> 
( [ z  /  x ] ph  ->  t  e.  z ) )  <->  ( (
z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
98albii 1470 . . . . . . . . 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 184 . . . . . . . 8  |-  ( A. z  e.  y  ( [ z  /  x ] ph  ->  t  e.  z )  <->  A. z
( ( z  e.  y  /\  [ z  /  x ] ph )  ->  t  e.  z ) )
11 sban 1955 . . . . . . . . . . . 12  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( [
z  /  x ]
x  e.  y  /\  [ z  /  x ] ph ) )
12 clelsb1 2282 . . . . . . . . . . . . 13  |-  ( [ z  /  x ]
x  e.  y  <->  z  e.  y )
1312anbi1i 458 . . . . . . . . . . . 12  |-  ( ( [ z  /  x ] x  e.  y  /\  [ z  /  x ] ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1411, 13bitri 184 . . . . . . . . . . 11  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  ( z  e.  y  /\  [ z  /  x ] ph ) )
1514bicomi 132 . . . . . . . . . 10  |-  ( ( z  e.  y  /\  [ z  /  x ] ph )  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
1615imbi1i 238 . . . . . . . . 9  |-  ( ( ( z  e.  y  /\  [ z  /  x ] ph )  -> 
t  e.  z )  <-> 
( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z ) )
1716albii 1470 . . . . . . . 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 184 . . . . . . 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 2164 . . . . . . . . . 10  |-  ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  <->  [ z  /  x ] ( x  e.  y  /\  ph )
)
2019bicomi 132 . . . . . . . . 9  |-  ( [ z  /  x ]
( x  e.  y  /\  ph )  <->  z  e.  { x  |  ( x  e.  y  /\  ph ) } )
2120imbi1i 238 . . . . . . . 8  |-  ( ( [ z  /  x ] ( x  e.  y  /\  ph )  ->  t  e.  z )  <-> 
( z  e.  {
x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) )
2221albii 1470 . . . . . . 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 184 . . . . . 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 14661 . . . . 5  |- BOUNDED  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z )
2524bdcab 14686 . . . 4  |- BOUNDED  { t  |  A. z ( z  e. 
{ x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
26 df-int 3847 . . . 4  |-  |^| { x  |  ( x  e.  y  /\  ph ) }  =  { t  |  A. z ( z  e.  { x  |  ( x  e.  y  /\  ph ) }  ->  t  e.  z ) }
2725, 26bdceqir 14681 . . 3  |- BOUNDED 
|^| { x  |  ( x  e.  y  /\  ph ) }
28 bdcriota.ex . . . . 5  |-  E! x  e.  y  ph
29 df-reu 2462 . . . . 5  |-  ( E! x  e.  y  ph  <->  E! x ( x  e.  y  /\  ph )
)
3028, 29mpbi 145 . . . 4  |-  E! x
( x  e.  y  /\  ph )
31 iotaint 5193 . . . 4  |-  ( E! x ( x  e.  y  /\  ph )  ->  ( iota x ( x  e.  y  /\  ph ) )  =  |^| { x  |  ( x  e.  y  /\  ph ) } )
3230, 31ax-mp 5 . . 3  |-  ( iota
x ( x  e.  y  /\  ph )
)  =  |^| { x  |  ( x  e.  y  /\  ph ) }
3327, 32bdceqir 14681 . 2  |- BOUNDED  ( iota x ( x  e.  y  /\  ph ) )
34 df-riota 5833 . 2  |-  ( iota_ x  e.  y  ph )  =  ( iota x
( x  e.  y  /\  ph ) )
3533, 34bdceqir 14681 1  |- BOUNDED  ( iota_ x  e.  y 
ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1351    = wceq 1353   [wsb 1762   E!weu 2026    e. wcel 2148   {cab 2163   A.wral 2455   E!wreu 2457   |^|cint 3846   iotacio 5178   iota_crio 5832  BOUNDED wbd 14649  BOUNDED wbdc 14677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14650  ax-bdim 14651  ax-bdal 14655  ax-bdel 14658  ax-bdsb 14659
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-iota 5180  df-riota 5833  df-bdc 14678
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator