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Theorem elmptrab2 17781
Description: Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
Hypotheses
Ref Expression
elmptrab2.f  |-  F  =  ( x  e.  _V  |->  { y  e.  B  |  ph } )
elmptrab2.s1  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
elmptrab2.s2  |-  ( x  =  X  ->  B  =  C )
elmptrab2.ex  |-  B  e.  V
elmptrab2.rc  |-  ( Y  e.  C  ->  X  e.  W )
Assertion
Ref Expression
elmptrab2  |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps )
)
Distinct variable groups:    x, y, ps    x, X, y    x, Y, y    x, C, y   
x, V, y    x, W, y    y, B
Allowed substitution hints:    ph( x, y)    B( x)    F( x, y)

Proof of Theorem elmptrab2
StepHypRef Expression
1 elmptrab2.f . . 3  |-  F  =  ( x  e.  _V  |->  { y  e.  B  |  ph } )
2 elmptrab2.s1 . . 3  |-  ( ( x  =  X  /\  y  =  Y )  ->  ( ph  <->  ps )
)
3 elmptrab2.s2 . . 3  |-  ( x  =  X  ->  B  =  C )
4 elmptrab2.ex . . . 4  |-  B  e.  V
54a1i 11 . . 3  |-  ( x  e.  _V  ->  B  e.  V )
61, 2, 3, 5elmptrab 17780 . 2  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
7 3simpc 956 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  ->  ( Y  e.  C  /\  ps ) )
8 elmptrab2.rc . . . . . 6  |-  ( Y  e.  C  ->  X  e.  W )
9 elex 2907 . . . . . 6  |-  ( X  e.  W  ->  X  e.  _V )
108, 9syl 16 . . . . 5  |-  ( Y  e.  C  ->  X  e.  _V )
1110adantr 452 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  X  e. 
_V )
12 simpl 444 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  Y  e.  C )
13 simpr 448 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  ps )
1411, 12, 133jca 1134 . . 3  |-  ( ( Y  e.  C  /\  ps )  ->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
157, 14impbii 181 . 2  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  <->  ( Y  e.  C  /\  ps )
)
166, 15bitri 241 1  |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    e. cmpt 4207   ` cfv 5394
This theorem is referenced by:  isfil  17800  isufil  17856
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402
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