Theorem bdcriota 15856

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.

Step	Hyp	Ref	Expression
1		bdcriota.bd	. . . . . . . . 9 |- BOUNDED ph
2	1	ax-bdsb 15795	. . . . . . . 8 |- BOUNDED [ z / x ] ph
3		ax-bdel 15794	. . . . . . . 8 |- BOUNDED t e. z
4	2, 3	ax-bdim 15787	. . . . . . 7 |- BOUNDED ( [ z / x ] ph -> t e. z )
5	4	ax-bdal 15791	. . . . . 6 |- BOUNDED A. z e. y ( [ z / x ] ph -> t e. z )
6		df-ral 2489	. . . . . . . . 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 ) ) )
8	7	bicomi 132	. . . . . . . . . 10 |- ( ( z e. y -> ( [ z / x ] ph -> t e. z ) ) <-> ( ( z e. y /\ [ z / x ] ph ) -> t e. z ) )
9	8	albii 1493	. . . . . . . . 9 |- ( A. z ( z e. y -> ( [ z / x ] ph -> t e. z ) ) <-> A. z ( ( z e. y /\ [ z / x ] ph ) -> t e. z ) )
10	6, 9	bitri 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 1983	. . . . . . . . . . . 12 |- ( [ z / x ] ( x e. y /\ ph ) <-> ( [ z / x ] x e. y /\ [ z / x ] ph ) )
12		clelsb1 2310	. . . . . . . . . . . . 13 |- ( [ z / x ] x e. y <-> z e. y )
13	12	anbi1i 458	. . . . . . . . . . . 12 |- ( ( [ z / x ] x e. y /\ [ z / x ] ph ) <-> ( z e. y /\ [ z / x ] ph ) )
14	11, 13	bitri 184	. . . . . . . . . . 11 |- ( [ z / x ] ( x e. y /\ ph ) <-> ( z e. y /\ [ z / x ] ph ) )
15	14	bicomi 132	. . . . . . . . . 10 |- ( ( z e. y /\ [ z / x ] ph ) <-> [ z / x ] ( x e. y /\ ph ) )
16	15	imbi1i 238	. . . . . . . . 9 |- ( ( ( z e. y /\ [ z / x ] ph ) -> t e. z ) <-> ( [ z / x ] ( x e. y /\ ph ) -> t e. z ) )
17	16	albii 1493	. . . . . . . 8 |- ( A. z ( ( z e. y /\ [ z / x ] ph ) -> t e. z ) <-> A. z ( [ z / x ] ( x e. y /\ ph ) -> t e. z ) )
18	10, 17	bitri 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 2192	. . . . . . . . . 10 |- ( z e. { x | ( x e. y /\ ph ) } <-> [ z / x ] ( x e. y /\ ph ) )
20	19	bicomi 132	. . . . . . . . 9 |- ( [ z / x ] ( x e. y /\ ph ) <-> z e. { x | ( x e. y /\ ph ) } )
21	20	imbi1i 238	. . . . . . . 8 |- ( ( [ z / x ] ( x e. y /\ ph ) -> t e. z ) <-> ( z e. { x | ( x e. y /\ ph ) } -> t e. z ) )
22	21	albii 1493	. . . . . . 7 |- ( A. z ( [ z / x ] ( x e. y /\ ph ) -> t e. z ) <-> A. z ( z e. { x | ( x e. y /\ ph ) } -> t e. z ) )
23	18, 22	bitri 184	. . . . . 6 |- ( A. z e. y ( [ z / x ] ph -> t e. z ) <-> A. z ( z e. { x | ( x e. y /\ ph ) } -> t e. z ) )
24	5, 23	bd0 15797	. . . . 5 |- BOUNDED A. z ( z e. { x | ( x e. y /\ ph ) } -> t e. z )
25	24	bdcab 15822	. . . 4 |- BOUNDED { t | A. z ( z e. { x | ( x e. y /\ ph ) } -> t e. z ) }
26		df-int 3886	. . . 4 |- |^| { x | ( x e. y /\ ph ) } = { t | A. z ( z e. { x | ( x e. y /\ ph ) } -> t e. z ) }
27	25, 26	bdceqir 15817	. . 3 |- BOUNDED |^| { x | ( x e. y /\ ph ) }
28		bdcriota.ex	. . . . 5 |- E! x e. y ph
29		df-reu 2491	. . . . 5 |- ( E! x e. y ph <-> E! x ( x e. y /\ ph ) )
30	28, 29	mpbi 145	. . . 4 |- E! x ( x e. y /\ ph )
31		iotaint 5246	. . . 4 |- ( E! x ( x e. y /\ ph ) -> ( iota x ( x e. y /\ ph ) ) = |^| { x | ( x e. y /\ ph ) } )
32	30, 31	ax-mp 5	. . 3 |- ( iota x ( x e. y /\ ph ) ) = |^| { x | ( x e. y /\ ph ) }
33	27, 32	bdceqir 15817	. 2 |- BOUNDED ( iota x ( x e. y /\ ph ) )
34		df-riota 5901	. 2 |- ( iota_ x e. y ph ) = ( iota x ( x e. y /\ ph ) )
35	33, 34	bdceqir 15817	1 |- BOUNDED ( iota_ x e. y ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   [wsb 1785   E!weu 2054   e. wcel 2176   {cab 2191   A.wral 2484   E!wreu 2486   |^|cint 3885   iotacio 5231   iota_crio 5900  BOUNDED wbd 15785  BOUNDED wbdc 15813

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15786  ax-bdim 15787  ax-bdal 15791  ax-bdel 15794  ax-bdsb 15795

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-iota 5233  df-riota 5901  df-bdc 15814

This theorem is referenced by: (None)