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Theorem riotasv2d 6365
Description: Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4556). Special case of riota2f 6342. (Contributed by NM, 2-Mar-2013.)
Hypotheses
Ref Expression
riotasv2d.1  |-  F/ y
ph
riotasv2d.2  |-  ( ph  -> 
F/_ y F )
riotasv2d.3  |-  ( ph  ->  F/ y ch )
riotasv2d.4  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
riotasv2d.5  |-  ( (
ph  /\  y  =  E )  ->  ( ps 
<->  ch ) )
riotasv2d.6  |-  ( (
ph  /\  y  =  E )  ->  C  =  F )
riotasv2d.7  |-  ( ph  ->  D  e.  A )
riotasv2d.8  |-  ( ph  ->  E  e.  B )
riotasv2d.9  |-  ( ph  ->  ch )
Assertion
Ref Expression
riotasv2d  |-  ( (
ph  /\  A  e.  V )  ->  D  =  F )
Distinct variable groups:    x, y, A    x, B, y    x, C    y, E    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x, y)    C( y)    D( x, y)    E( x)    F( x, y)    V( x, y)

Proof of Theorem riotasv2d
StepHypRef Expression
1 elex 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv2d.8 . . . 4  |-  ( ph  ->  E  e.  B )
32adantr 451 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E  e.  B )
4 riotasv2d.9 . . . 4  |-  ( ph  ->  ch )
54adantr 451 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ch )
6 eleq1 2356 . . . . . . . 8  |-  ( y  =  E  ->  (
y  e.  B  <->  E  e.  B ) )
76adantl 452 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  (
y  e.  B  <->  E  e.  B ) )
8 riotasv2d.5 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  ( ps 
<->  ch ) )
97, 8anbi12d 691 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  (
( y  e.  B  /\  ps )  <->  ( E  e.  B  /\  ch )
) )
10 riotasv2d.6 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  C  =  F )
1110eqeq2d 2307 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  ( D  =  C  <->  D  =  F ) )
129, 11imbi12d 311 . . . . 5  |-  ( (
ph  /\  y  =  E )  ->  (
( ( y  e.  B  /\  ps )  ->  D  =  C )  <-> 
( ( E  e.  B  /\  ch )  ->  D  =  F ) ) )
1312adantlr 695 . . . 4  |-  ( ( ( ph  /\  A  e.  _V )  /\  y  =  E )  ->  (
( ( y  e.  B  /\  ps )  ->  D  =  C )  <-> 
( ( E  e.  B  /\  ch )  ->  D  =  F ) ) )
14 riotasv2d.4 . . . . 5  |-  ( ph  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) )
15 riotasv2d.7 . . . . 5  |-  ( ph  ->  D  e.  A )
1614, 15riotasvd 6363 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
17 riotasv2d.1 . . . . 5  |-  F/ y
ph
18 nfv 1609 . . . . 5  |-  F/ y  A  e.  _V
1917, 18nfan 1783 . . . 4  |-  F/ y ( ph  /\  A  e.  _V )
20 nfcvd 2433 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/_ y E )
21 nfvd 1610 . . . . . . 7  |-  ( ph  ->  F/ y  E  e.  B )
22 riotasv2d.3 . . . . . . 7  |-  ( ph  ->  F/ y ch )
2321, 22nfand 1775 . . . . . 6  |-  ( ph  ->  F/ y ( E  e.  B  /\  ch ) )
24 nfra1 2606 . . . . . . . . 9  |-  F/ y A. y  e.  B  ( ps  ->  x  =  C )
25 nfcv 2432 . . . . . . . . 9  |-  F/_ y A
2624, 25nfriota 6330 . . . . . . . 8  |-  F/_ y
( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) )
2717, 14nfceqdf 2431 . . . . . . . 8  |-  ( ph  ->  ( F/_ y D  <->  F/_ y ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) ) )
2826, 27mpbiri 224 . . . . . . 7  |-  ( ph  -> 
F/_ y D )
29 riotasv2d.2 . . . . . . 7  |-  ( ph  -> 
F/_ y F )
3028, 29nfeqd 2446 . . . . . 6  |-  ( ph  ->  F/ y  D  =  F )
3123, 30nfimd 1773 . . . . 5  |-  ( ph  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
3231adantr 451 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
333, 13, 16, 19, 20, 32vtocldf 2848 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
343, 5, 33mp2and 660 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  D  =  F )
351, 34sylan2 460 1  |-  ( (
ph  /\  A  e.  V )  ->  D  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   A.wral 2556   _Vcvv 2801   iota_crio 6313
This theorem is referenced by:  riotasv2s  6367  cdleme42b  31289
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
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