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Theorem riotasv 32595
Description: Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4607). Special case of riota2f 6291. (Contributed by NM, 26-Jan-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
riotasv.1  |-  A  e. 
_V
riotasv.2  |-  D  =  ( iota_ x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
Assertion
Ref Expression
riotasv  |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
Distinct variable groups:    x, y, A    x, B    x, C    ph, x
Allowed substitution hints:    ph( y)    B( y)    C( y)    D( x, y)

Proof of Theorem riotasv
StepHypRef Expression
1 riotasv.1 . . 3  |-  A  e. 
_V
2 riotasv.2 . . . . 5  |-  D  =  ( iota_ x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) )
32a1i 11 . . . 4  |-  ( D  e.  A  ->  D  =  ( iota_ x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) ) )
4 id 22 . . . 4  |-  ( D  e.  A  ->  D  e.  A )
53, 4riotasvd 32592 . . 3  |-  ( ( D  e.  A  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ph )  ->  D  =  C ) )
61, 5mpan2 685 . 2  |-  ( D  e.  A  ->  (
( y  e.  B  /\  ph )  ->  D  =  C ) )
763impib 1229 1  |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031   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-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:  cdleme26e  33997  cdleme26eALTN  33999  cdleme26fALTN  34000  cdleme26f  34001  cdleme26f2ALTN  34002  cdleme26f2  34003
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