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

Theorem riotasv2s 6493
Description: The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4643) in the form of a substitution instance. Special case of riota2f 6468. (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 955 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  -> 
( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) ) )
2 simp1 956 . 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 2678 . . . . . . 7  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
5 nfcv 2502 . . . . . . 7  |-  F/_ y A
64, 5nfriota 6456 . . . . . 6  |-  F/_ y
( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )
73, 6nfcxfr 2499 . . . . 5  |-  F/_ y D
87nfel1 2512 . . . 4  |-  F/ y  D  e.  A
9 nfv 1624 . . . . 5  |-  F/ y  E  e.  B
10 nfsbc1v 3096 . . . . 5  |-  F/ y
[. E  /  y ]. ph
119, 10nfan 1834 . . . 4  |-  F/ y ( E  e.  B  /\  [. E  /  y ]. ph )
128, 11nfan 1834 . . 3  |-  F/ y ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )
13 nfcsb1v 3199 . . . 4  |-  F/_ y [_ E  /  y ]_ C
1413a1i 10 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  F/_ y [_ E  / 
y ]_ C )
1510a1i 10 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  F/ y [. E  / 
y ]. ph )
163a1i 10 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) ) )
17 sbceq1a 3087 . . . 4  |-  ( y  =  E  ->  ( ph 
<-> 
[. E  /  y ]. ph ) )
1817adantl 452 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  ( ph  <->  [. E  / 
y ]. ph ) )
19 csbeq1a 3175 . . . 4  |-  ( y  =  E  ->  C  =  [_ E  /  y ]_ C )
2019adantl 452 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  C  =  [_ E  /  y ]_ C
)
21 simpl 443 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  e.  A )
22 simprl 732 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  E  e.  B )
23 simprr 733 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  [. E  /  y ]. ph )
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 6491 . 2  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  A  e.  V )  ->  D  =  [_ E  /  y ]_ C
)
251, 2, 24syl2anc 642 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 176    /\ wa 358    /\ w3a 935   F/wnf 1549    = wceq 1647    e. wcel 1715   F/_wnfc 2489   A.wral 2628   [.wsbc 3077   [_csb 3167   iota_crio 6439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-undef 6440  df-riota 6446
  Copyright terms: Public domain W3C validator