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Theorem elmptrab2 20156
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 20155 . 2  |-  ( Y  e.  ( F `  X )  <->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
7 3simpc 995 . . 3  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  ->  ( Y  e.  C  /\  ps ) )
8 elmptrab2.rc . . . . . 6  |-  ( Y  e.  C  ->  X  e.  W )
9 elex 3122 . . . . . 6  |-  ( X  e.  W  ->  X  e.  _V )
108, 9syl 16 . . . . 5  |-  ( Y  e.  C  ->  X  e.  _V )
1110adantr 465 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  X  e. 
_V )
12 simpl 457 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  Y  e.  C )
13 simpr 461 . . . 4  |-  ( ( Y  e.  C  /\  ps )  ->  ps )
1411, 12, 133jca 1176 . . 3  |-  ( ( Y  e.  C  /\  ps )  ->  ( X  e.  _V  /\  Y  e.  C  /\  ps )
)
157, 14impbii 188 . 2  |-  ( ( X  e.  _V  /\  Y  e.  C  /\  ps )  <->  ( Y  e.  C  /\  ps )
)
166, 15bitri 249 1  |-  ( Y  e.  ( F `  X )  <->  ( Y  e.  C  /\  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    |-> cmpt 4505   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  isfil  20175  isufil  20231
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