MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotasv3d Unicode version

Theorem riotasv3d 6369
Description: A property  ch holding for a representative of a single-valued class expression  C ( y ) (see e.g. reusv2 4556) also holds for its description binder  D (in the form of property  th). (Contributed by NM, 5-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotasv3d.1  |-  F/ y
ph
riotasv3d.2  |-  ( ph  ->  F/ y th )
riotasv3d.3  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
riotasv3d.4  |-  ( (
ph  /\  C  =  D )  ->  ( ch 
<->  th ) )
riotasv3d.5  |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )
riotasv3d.6  |-  ( ph  ->  D  e.  A )
riotasv3d.7  |-  ( ph  ->  E. y  e.  B  ps )
Assertion
Ref Expression
riotasv3d  |-  ( (
ph  /\  A  e.  V )  ->  th )
Distinct variable groups:    x, y, A    x, B    x, C    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x, y)    th( x, y)    B( y)    C( y)    D( x, y)    V( x, y)

Proof of Theorem riotasv3d
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv3d.7 . . . 4  |-  ( ph  ->  E. y  e.  B  ps )
32adantr 451 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E. y  e.  B  ps )
4 riotasv3d.1 . . . . . 6  |-  F/ y
ph
5 nfv 1609 . . . . . 6  |-  F/ y  A  e.  _V
6 riotasv3d.5 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  e.  B  /\  ps )  ->  ch ) )
76imp 418 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  B  /\  ps )
)  ->  ch )
87adantrl 696 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  ch )
9 riotasv3d.3 . . . . . . . . . . . 12  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
10 riotasv3d.6 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  A )
119, 10riotasvd 6363 . . . . . . . . . . 11  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
1211impr 602 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  D  =  C )
1312eqcomd 2301 . . . . . . . . 9  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  C  =  D )
14 riotasv3d.4 . . . . . . . . 9  |-  ( (
ph  /\  C  =  D )  ->  ( ch 
<->  th ) )
1513, 14syldan 456 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  -> 
( ch  <->  th )
)
168, 15mpbid 201 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  _V  /\  ( y  e.  B  /\  ps ) ) )  ->  th )
1716exp45 597 . . . . . 6  |-  ( ph  ->  ( A  e.  _V  ->  ( y  e.  B  ->  ( ps  ->  th )
) ) )
184, 5, 17ralrimd 2644 . . . . 5  |-  ( ph  ->  ( A  e.  _V  ->  A. y  e.  B  ( ps  ->  th )
) )
19 riotasv3d.2 . . . . . 6  |-  ( ph  ->  F/ y th )
20 r19.23t 2670 . . . . . 6  |-  ( F/ y th  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2119, 20syl 15 . . . . 5  |-  ( ph  ->  ( A. y  e.  B  ( ps  ->  th )  <->  ( E. y  e.  B  ps  ->  th ) ) )
2218, 21sylibd 205 . . . 4  |-  ( ph  ->  ( A  e.  _V  ->  ( E. y  e.  B  ps  ->  th )
) )
2322imp 418 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( E. y  e.  B  ps  ->  th ) )
243, 23mpd 14 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  th )
251, 24sylan2 460 1  |-  ( (
ph  /\  A  e.  V )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   iota_crio 6313
This theorem is referenced by:  cdlemefs32sn1aw  31225  cdleme43fsv1snlem  31231  cdleme41sn3a  31244  cdleme40m  31278  cdleme40n  31279  cdlemkid  31747  dihvalcqpre  32047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-undef 6314  df-riota 6320
  Copyright terms: Public domain W3C validator