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Theorem elmptrab2 17539
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 10 . . 3  |-  ( x  e.  _V  ->  B  e.  V )
61, 2, 3, 5elmptrab 17538 . 2  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
7 3simpc 954 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  ->  ( Y  e.  C  /\  ps ) )
8 elmptrab2.rc . . . . . 6  |-  ( Y  e.  C  ->  X  e.  W )
9 elex 2809 . . . . . 6  |-  ( X  e.  W  ->  X  e.  _V )
108, 9syl 15 . . . . 5  |-  ( Y  e.  C  ->  X  e.  _V )
1110adantr 451 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  X  e. 
_V )
12 simpl 443 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  Y  e.  C )
13 simpr 447 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  ps )
1411, 12, 133jca 1132 . . 3  |-  ( ( Y  e.  C  /\  ps )  ->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
157, 14impbii 180 . 2  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  <->  ( Y  e.  C  /\  ps )
)
166, 15bitri 240 1  |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  isfil  17558  isufil  17614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279
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