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Theorem riotasv2s 6351
Description: The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4540) in the form of a substitution instance. Special case of riota2f 6326. (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 954 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  -> 
( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) ) )
2 simp1 955 . 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 2593 . . . . . . 7  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
5 nfcv 2419 . . . . . . 7  |-  F/_ y A
64, 5nfriota 6314 . . . . . 6  |-  F/_ y
( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) )
73, 6nfcxfr 2416 . . . . 5  |-  F/_ y D
87nfel1 2429 . . . 4  |-  F/ y  D  e.  A
9 nfv 1605 . . . . 5  |-  F/ y  E  e.  B
10 nfsbc1v 3010 . . . . 5  |-  F/ y
[. E  /  y ]. ph
119, 10nfan 1771 . . . 4  |-  F/ y ( E  e.  B  /\  [. E  /  y ]. ph )
128, 11nfan 1771 . . 3  |-  F/ y ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )
13 nfcsb1v 3113 . . . 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 3001 . . . 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 3089 . . . 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 6349 . 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 934   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   A.wral 2543   [.wsbc 2991   [_csb 3081   iota_crio 6297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-undef 6298  df-riota 6304
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