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Theorem riotasv2s 32594
Description: The value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4607) in the form of a substitution instance. Special case of riota2f 6291. (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 1029 . 2  |-  ( ( A  e.  V  /\  D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  -> 
( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) ) )
2 simp1 1030 . 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 2785 . . . . . . 7  |-  F/ y A. y  e.  B  ( ph  ->  x  =  C )
5 nfcv 2612 . . . . . . 7  |-  F/_ y A
64, 5nfriota 6279 . . . . . 6  |-  F/_ y
( iota_ x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
73, 6nfcxfr 2610 . . . . 5  |-  F/_ y D
87nfel1 2626 . . . 4  |-  F/ y  D  e.  A
9 nfv 1769 . . . . 5  |-  F/ y  E  e.  B
10 nfsbc1v 3275 . . . . 5  |-  F/ y
[. E  /  y ]. ph
119, 10nfan 2031 . . . 4  |-  F/ y ( E  e.  B  /\  [. E  /  y ]. ph )
128, 11nfan 2031 . . 3  |-  F/ y ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )
13 nfcsb1v 3365 . . . 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 3266 . . . 4  |-  ( y  =  E  ->  ( ph 
<-> 
[. E  /  y ]. ph ) )
1817adantl 473 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  ( ph  <->  [. E  / 
y ]. ph ) )
19 csbeq1a 3358 . . . 4  |-  ( y  =  E  ->  C  =  [_ E  /  y ]_ C )
2019adantl 473 . . 3  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  y  =  E )  ->  C  =  [_ E  /  y ]_ C
)
21 simpl 464 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  D  e.  A )
22 simprl 772 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  E  e.  B )
23 simprr 774 . . 3  |-  ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  ->  [. E  /  y ]. ph )
2412, 14, 15, 16, 18, 20, 21, 22, 23riotasv2d 32593 . 2  |-  ( ( ( D  e.  A  /\  ( E  e.  B  /\  [. E  /  y ]. ph ) )  /\  A  e.  V )  ->  D  =  [_ E  /  y ]_ C
)
251, 2, 24syl2anc 673 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452   F/wnf 1675    e. wcel 1904   F/_wnfc 2599   A.wral 2756   [.wsbc 3255   [_csb 3349   iota_crio 6269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-riota 6270  df-undef 7038
This theorem is referenced by: (None)
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