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Theorem elmptrab2 17852
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 17851 . 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 2956 . . . . . 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    e. cmpt 4258   ` cfv 5446
This theorem is referenced by:  isfil  17871  isufil  17927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454
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