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Theorem riotaxfrd 6581
Description: Change the variable  x in the expression for "the unique  x such that  ps " to another variable  y contained in expression  B. Use reuhypd 4750 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotaxfrd.1  |-  F/_ y C
riotaxfrd.2  |-  ( (
ph  /\  y  e.  A )  ->  B  e.  A )
riotaxfrd.3  |-  ( (
ph  /\  ( iota_ y  e.  A ch )  e.  A )  ->  C  e.  A )
riotaxfrd.4  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
riotaxfrd.5  |-  ( y  =  ( iota_ y  e.  A ch )  ->  B  =  C )
riotaxfrd.6  |-  ( (
ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )
Assertion
Ref Expression
riotaxfrd  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  C
)
Distinct variable groups:    x, B    x, C    x, y, A    ph, x, y    ps, y    ch, x
Allowed substitution hints:    ps( x)    ch( y)    B( y)    C( y)

Proof of Theorem riotaxfrd
StepHypRef Expression
1 rabid 2884 . . . 4  |-  ( x  e.  { x  e.  A  |  ps }  <->  ( x  e.  A  /\  ps ) )
21baib 872 . . 3  |-  ( x  e.  A  ->  (
x  e.  { x  e.  A  |  ps } 
<->  ps ) )
32riotabiia 6567 . 2  |-  ( iota_ x  e.  A x  e. 
{ x  e.  A  |  ps } )  =  ( iota_ x  e.  A ps )
4 riotaxfrd.2 . . . . . 6  |-  ( (
ph  /\  y  e.  A )  ->  B  e.  A )
5 riotaxfrd.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  E! y  e.  A  x  =  B )
6 riotaxfrd.4 . . . . . 6  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
74, 5, 6reuxfrd 4748 . . . . 5  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! y  e.  A  ch )
)
8 riotacl2 6563 . . . . . . . 8  |-  ( E! y  e.  A  ch  ->  ( iota_ y  e.  A ch )  e.  { y  e.  A  |  ch } )
98adantl 453 . . . . . . 7  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  ( iota_ y  e.  A ch )  e.  { y  e.  A  |  ch } )
10 riotacl 6564 . . . . . . . 8  |-  ( E! y  e.  A  ch  ->  ( iota_ y  e.  A ch )  e.  A
)
11 nfriota1 6557 . . . . . . . . 9  |-  F/_ y
( iota_ y  e.  A ch )
12 riotaxfrd.1 . . . . . . . . 9  |-  F/_ y C
13 riotaxfrd.5 . . . . . . . . 9  |-  ( y  =  ( iota_ y  e.  A ch )  ->  B  =  C )
1411, 12, 4, 6, 13rabxfrd 4744 . . . . . . . 8  |-  ( (
ph  /\  ( iota_ y  e.  A ch )  e.  A )  ->  ( C  e.  { x  e.  A  |  ps } 
<->  ( iota_ y  e.  A ch )  e.  { y  e.  A  |  ch } ) )
1510, 14sylan2 461 . . . . . . 7  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  ( C  e.  {
x  e.  A  |  ps }  <->  ( iota_ y  e.  A ch )  e. 
{ y  e.  A  |  ch } ) )
169, 15mpbird 224 . . . . . 6  |-  ( (
ph  /\  E! y  e.  A  ch )  ->  C  e.  { x  e.  A  |  ps } )
1716ex 424 . . . . 5  |-  ( ph  ->  ( E! y  e.  A  ch  ->  C  e.  { x  e.  A  |  ps } ) )
187, 17sylbid 207 . . . 4  |-  ( ph  ->  ( E! x  e.  A  ps  ->  C  e.  { x  e.  A  |  ps } ) )
1918imp 419 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  C  e.  { x  e.  A  |  ps } )
20 riotaxfrd.3 . . . . . . . 8  |-  ( (
ph  /\  ( iota_ y  e.  A ch )  e.  A )  ->  C  e.  A )
2120ex 424 . . . . . . 7  |-  ( ph  ->  ( ( iota_ y  e.  A ch )  e.  A  ->  C  e.  A ) )
2210, 21syl5 30 . . . . . 6  |-  ( ph  ->  ( E! y  e.  A  ch  ->  C  e.  A ) )
237, 22sylbid 207 . . . . 5  |-  ( ph  ->  ( E! x  e.  A  ps  ->  C  e.  A ) )
2423imp 419 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  C  e.  A )
251baibr 873 . . . . . . 7  |-  ( x  e.  A  ->  ( ps 
<->  x  e.  { x  e.  A  |  ps } ) )
2625reubiia 2893 . . . . . 6  |-  ( E! x  e.  A  ps  <->  E! x  e.  A  x  e.  { x  e.  A  |  ps }
)
2726biimpi 187 . . . . 5  |-  ( E! x  e.  A  ps  ->  E! x  e.  A  x  e.  { x  e.  A  |  ps } )
2827adantl 453 . . . 4  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  x  e.  { x  e.  A  |  ps } )
29 nfcv 2572 . . . . 5  |-  F/_ x C
30 nfrab1 2888 . . . . . 6  |-  F/_ x { x  e.  A  |  ps }
3130nfel2 2584 . . . . 5  |-  F/ x  C  e.  { x  e.  A  |  ps }
32 eleq1 2496 . . . . 5  |-  ( x  =  C  ->  (
x  e.  { x  e.  A  |  ps } 
<->  C  e.  { x  e.  A  |  ps } ) )
3329, 31, 32riota2f 6571 . . . 4  |-  ( ( C  e.  A  /\  E! x  e.  A  x  e.  { x  e.  A  |  ps } )  ->  ( C  e.  { x  e.  A  |  ps } 
<->  ( iota_ x  e.  A x  e.  { x  e.  A  |  ps } )  =  C ) )
3424, 28, 33syl2anc 643 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( C  e.  {
x  e.  A  |  ps }  <->  ( iota_ x  e.  A x  e.  {
x  e.  A  |  ps } )  =  C ) )
3519, 34mpbid 202 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A x  e.  { x  e.  A  |  ps } )  =  C )
363, 35syl5eqr 2482 1  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  ( iota_ x  e.  A ps )  =  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   F/_wnfc 2559   E!wreu 2707   {crab 2709   iota_crio 6542
This theorem is referenced by:  riotaneg  9983  riotaocN  30007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-riota 6549
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