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Theorem riotasv 6352
Description: Value of description binder  D for a single-valued class expression  C ( y ) (as in e.g. reusv2 4540). Special case of riota2f 6326. (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 10 . . . 4  |-  ( D  e.  A  ->  D  =  ( iota_ x  e.  A A. y  e.  B  ( ph  ->  x  =  C ) ) )
4 id 19 . . . 4  |-  ( D  e.  A  ->  D  e.  A )
53, 4riotasvd 6347 . . 3  |-  ( ( D  e.  A  /\  A  e.  _V )  ->  ( ( y  e.  B  /\  ph )  ->  D  =  C ) )
61, 5mpan2 652 . 2  |-  ( D  e.  A  ->  (
( y  e.  B  /\  ph )  ->  D  =  C ) )
763impib 1149 1  |-  ( ( D  e.  A  /\  y  e.  B  /\  ph )  ->  D  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   iota_crio 6297
This theorem is referenced by:  cdleme26e  30548  cdleme26eALTN  30550  cdleme26fALTN  30551  cdleme26f  30552  cdleme26f2ALTN  30553  cdleme26f2  30554
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-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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