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Theorem elmptrab2 17523
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 17522 . 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 2796 . . . . . 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    e. cmpt 4077   ` cfv 5255
This theorem is referenced by:  isfil  17542  isufil  17598
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
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263
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