MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotasv2s Unicode version

Theorem riotasv2s 6563
Description: The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4696) in the form of a substitution instance. Special case of riota2f 6538. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypothesis
Ref Expression
riotasv2s.2  |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )
Assertion
Ref Expression
riotasv2s  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  / 
y ]_ C )
Distinct variable groups:    x, y, A    x, B, y    x, C    x, E, y    ph, x
Allowed substitution hints:    ph( y)    C( y)    D( x, y)    V( x, y)

Proof of Theorem riotasv2s
StepHypRef Expression
1 3simpc 956 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  -> 
( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) ) )
2 simp1 957 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  A  e.  V )
3 riotasv2s.2 . . . . . 6  |-  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )
4 nfra1 2724 . . . . . . 7  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
5 nfcv 2548 . . . . . . 7  |-  F/_ y A
64, 5nfriota 6526 . . . . . 6  |-  F/_ y
( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )
73, 6nfcxfr 2545 . . . . 5  |-  F/_ y D
87nfel1 2558 . . . 4  |-  F/ y  D  e.  A
9 nfv 1626 . . . . 5  |-  F/ y  E  e.  B
10 nfsbc1v 3148 . . . . 5  |-  F/ y
[. E  /  y ]. ph
119, 10nfan 1842 . . . 4  |-  F/ y ( E  e.  B  /\  [. E  /  y ]. ph )
128, 11nfan 1842 . . 3  |-  F/ y ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )
13 nfcsb1v 3251 . . . 4  |-  F/_ y [_ E  /  y ]_ C
1413a1i 11 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  F/_ y [_ E  / 
y ]_ C )
1510a1i 11 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  F/ y [. E  / 
y ]. ph )
163a1i 11 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) ) )
17 sbceq1a 3139 . . . 4  |-  ( y  =  E  ->  ( ph 
<-> 
[. E  /  y ]. ph ) )
1817adantl 453 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  ( ph  <->  [. E  / 
y ]. ph ) )
19 csbeq1a 3227 . . . 4  |-  ( y  =  E  ->  C  =  [_ E  /  y ]_ C )
2019adantl 453 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  C  =  [_ E  /  y ]_ C
)
21 simpl 444 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  e.  A )
22 simprl 733 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  E  e.  B )
23 simprr 734 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  [. E  /  y ]. ph )
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 6561 . 2  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  A  e.  V )  ->  D  =  [_ E  /  y ]_ C
)
251, 2, 24syl2anc 643 1  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  / 
y ]_ C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2535   A.wral 2674   [.wsbc 3129   [_csb 3219   iota_crio 6509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-undef 6510  df-riota 6516
  Copyright terms: Public domain W3C validator