MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotasv2d Structured version   Unicode version

Theorem riotasv2d 6586
Description: Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4721). Special case of riota2f 6563. (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 2956 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 riotasv2d.8 . . . 4  |-  ( ph  ->  E  e.  B )
32adantr 452 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  E  e.  B )
4 riotasv2d.9 . . . 4  |-  ( ph  ->  ch )
54adantr 452 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ch )
6 eleq1 2495 . . . . . . . 8  |-  ( y  =  E  ->  (
y  e.  B  <->  E  e.  B ) )
76adantl 453 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  (
y  e.  B  <->  E  e.  B ) )
8 riotasv2d.5 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  ( ps 
<->  ch ) )
97, 8anbi12d 692 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  (
( y  e.  B  /\  ps )  <->  ( E  e.  B  /\  ch )
) )
10 riotasv2d.6 . . . . . . 7  |-  ( (
ph  /\  y  =  E )  ->  C  =  F )
1110eqeq2d 2446 . . . . . 6  |-  ( (
ph  /\  y  =  E )  ->  ( D  =  C  <->  D  =  F ) )
129, 11imbi12d 312 . . . . 5  |-  ( (
ph  /\  y  =  E )  ->  (
( ( y  e.  B  /\  ps )  ->  D  =  C )  <-> 
( ( E  e.  B  /\  ch )  ->  D  =  F ) ) )
1312adantlr 696 . . . 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 6584 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ps )  ->  D  =  C ) )
17 riotasv2d.1 . . . . 5  |-  F/ y
ph
18 nfv 1629 . . . . 5  |-  F/ y  A  e.  _V
1917, 18nfan 1846 . . . 4  |-  F/ y ( ph  /\  A  e.  _V )
20 nfcvd 2572 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/_ y E )
21 nfvd 1630 . . . . . . 7  |-  ( ph  ->  F/ y  E  e.  B )
22 riotasv2d.3 . . . . . . 7  |-  ( ph  ->  F/ y ch )
2321, 22nfand 1843 . . . . . 6  |-  ( ph  ->  F/ y ( E  e.  B  /\  ch ) )
24 nfra1 2748 . . . . . . . . 9  |-  F/ y A. y  e.  B  ( ps  ->  x  =  C )
25 nfcv 2571 . . . . . . . . 9  |-  F/_ y A
2624, 25nfriota 6551 . . . . . . . 8  |-  F/_ y
( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) )
2717, 14nfceqdf 2570 . . . . . . . 8  |-  ( ph  ->  ( F/_ y D  <->  F/_ y ( iota_ x  e.  A A. y  e.  B  ( ps  ->  x  =  C ) ) ) )
2826, 27mpbiri 225 . . . . . . 7  |-  ( ph  -> 
F/_ y D )
29 riotasv2d.2 . . . . . . 7  |-  ( ph  -> 
F/_ y F )
3028, 29nfeqd 2585 . . . . . 6  |-  ( ph  ->  F/ y  D  =  F )
3123, 30nfimd 1827 . . . . 5  |-  ( ph  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
3231adantr 452 . . . 4  |-  ( (
ph  /\  A  e.  _V )  ->  F/ y ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
333, 13, 16, 19, 20, 32vtocldf 2995 . . 3  |-  ( (
ph  /\  A  e.  _V )  ->  ( ( E  e.  B  /\  ch )  ->  D  =  F ) )
343, 5, 33mp2and 661 . 2  |-  ( (
ph  /\  A  e.  _V )  ->  D  =  F )
351, 34sylan2 461 1  |-  ( (
ph  /\  A  e.  V )  ->  D  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   F/_wnfc 2558   A.wral 2697   _Vcvv 2948   iota_crio 6534
This theorem is referenced by:  riotasv2s  6588  cdleme42b  31176
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-undef 6535  df-riota 6541
  Copyright terms: Public domain W3C validator