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Theorem riotasv2d 34831
Description: Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4662). Special case of riota2f 6279. (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 3118 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv2d.8 . . . 4  |-  ( ph  ->  E  e.  B )
32adantr 465 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E  e.  B )
4 riotasv2d.9 . . . 4  |-  ( ph  ->  ch )
54adantr 465 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ch )
6 eleq1 2529 . . . . . . . 8  |-  ( y  =  E  ->  (
y  e.  B  <->  E  e.  B ) )
76adantl 466 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  (
y  e.  B  <->  E  e.  B ) )
8 riotasv2d.5 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  ( ps 
<->  ch ) )
97, 8anbi12d 710 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  (
( y  e.  B  /\  ps )  <->  ( E  e.  B  /\  ch )
) )
10 riotasv2d.6 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  C  =  F )
1110eqeq2d 2471 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  ( D  =  C  <->  D  =  F ) )
129, 11imbi12d 320 . . . . 5  |-  ( (
ph  /\  y  =  E )  ->  (
( ( y  e.  B  /\  ps )  ->  D  =  C )  <-> 
( ( E  e.  B  /\  ch )  ->  D  =  F ) ) )
1312adantlr 714 . . . 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 34830 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
17 riotasv2d.1 . . . . 5  |-  F/ y
ph
18 nfv 1708 . . . . 5  |-  F/ y  A  e.  _V
1917, 18nfan 1929 . . . 4  |-  F/ y ( ph  /\  A  e.  _V )
20 nfcvd 2620 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/_ y E )
21 nfvd 1709 . . . . . . 7  |-  ( ph  ->  F/ y  E  e.  B )
22 riotasv2d.3 . . . . . . 7  |-  ( ph  ->  F/ y ch )
2321, 22nfand 1926 . . . . . 6  |-  ( ph  ->  F/ y ( E  e.  B  /\  ch ) )
24 nfra1 2838 . . . . . . . . 9  |-  F/ y A. y  e.  B  ( ps  ->  x  =  C )
25 nfcv 2619 . . . . . . . . 9  |-  F/_ y A
2624, 25nfriota 6267 . . . . . . . 8  |-  F/_ y
( iota_ x  e.  A  A. y  e.  B  ( ps  ->  x  =  C ) )
2717, 14nfceqdf 2614 . . . . . . . 8  |-  ( ph  ->  ( F/_ y D  <->  F/_ y ( iota_ x  e.  A  A. y  e.  B  ( ps  ->  x  =  C ) ) ) )
2826, 27mpbiri 233 . . . . . . 7  |-  ( ph  -> 
F/_ y D )
29 riotasv2d.2 . . . . . . 7  |-  ( ph  -> 
F/_ y F )
3028, 29nfeqd 2626 . . . . . 6  |-  ( ph  ->  F/ y  D  =  F )
3123, 30nfimd 1918 . . . . 5  |-  ( ph  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
3231adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
333, 13, 16, 19, 20, 32vtocldf 3158 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
343, 5, 33mp2and 679 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  D  =  F )
351, 34sylan2 474 1  |-  ( (
ph  /\  A  e.  V )  ->  D  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395   F/wnf 1617    e. wcel 1819   F/_wnfc 2605   A.wral 2807   _Vcvv 3109   iota_crio 6257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-riotaBAD 34827
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-riota 6258  df-undef 7020
This theorem is referenced by:  riotasv2s  34832  cdleme42b  36347
  Copyright terms: Public domain W3C validator