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Theorem riotasv2s 33761
Description: The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4653) in the form of a substitution instance. Special case of riota2f 6265. (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 995 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  -> 
( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) ) )
2 simp1 996 . 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 2845 . . . . . . 7  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
5 nfcv 2629 . . . . . . 7  |-  F/_ y A
64, 5nfriota 6252 . . . . . 6  |-  F/_ y
( iota_ x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
73, 6nfcxfr 2627 . . . . 5  |-  F/_ y D
87nfel1 2645 . . . 4  |-  F/ y  D  e.  A
9 nfv 1683 . . . . 5  |-  F/ y  E  e.  B
10 nfsbc1v 3351 . . . . 5  |-  F/ y
[. E  /  y ]. ph
119, 10nfan 1875 . . . 4  |-  F/ y ( E  e.  B  /\  [. E  /  y ]. ph )
128, 11nfan 1875 . . 3  |-  F/ y ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )
13 nfcsb1v 3451 . . . 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 3342 . . . 4  |-  ( y  =  E  ->  ( ph 
<-> 
[. E  /  y ]. ph ) )
1817adantl 466 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  ( ph  <->  [. E  / 
y ]. ph ) )
19 csbeq1a 3444 . . . 4  |-  ( y  =  E  ->  C  =  [_ E  /  y ]_ C )
2019adantl 466 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  C  =  [_ E  /  y ]_ C
)
21 simpl 457 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  e.  A )
22 simprl 755 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  E  e.  B )
23 simprr 756 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  [. E  /  y ]. ph )
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 33760 . 2  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  A  e.  V )  ->  D  =  [_ E  /  y ]_ C
)
251, 2, 24syl2anc 661 1  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  =  [_ E  / 
y ]_ C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   A.wral 2814   [.wsbc 3331   [_csb 3435   iota_crio 6242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-riota 6243  df-undef 6999
This theorem is referenced by: (None)
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