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Theorem abfmpel 23911
Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
Hypotheses
Ref Expression
abfmpel.1  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
abfmpel.2  |-  { y  |  ph }  e.  _V
abfmpel.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
abfmpel  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    x, V, y   
y, W    ps, x, y
Allowed substitution hints:    ph( x, y)    W( x)

Proof of Theorem abfmpel
StepHypRef Expression
1 abfmpel.2 . . . . . . . 8  |-  { y  |  ph }  e.  _V
21ax-gen 1552 . . . . . . 7  |-  A. x { y  |  ph }  e.  _V
3 csbexg 3206 . . . . . . 7  |-  ( ( A  e.  V  /\  A. x { y  | 
ph }  e.  _V )  ->  [_ A  /  x ]_ { y  |  ph }  e.  _V )
42, 3mpan2 653 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  e.  _V )
5 abfmpel.1 . . . . . . 7  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
65fvmpts 5748 . . . . . 6  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ph }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ph } )
74, 6mpdan 650 . . . . 5  |-  ( A  e.  V  ->  ( F `  A )  =  [_ A  /  x ]_ { y  |  ph } )
8 csbabg 3255 . . . . 5  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
97, 8eqtrd 2421 . . . 4  |-  ( A  e.  V  ->  ( F `  A )  =  { y  |  [. A  /  x ]. ph }
)
109eleq2d 2456 . . 3  |-  ( A  e.  V  ->  ( B  e.  ( F `  A )  <->  B  e.  { y  |  [. A  /  x ]. ph }
) )
1110adantr 452 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ph } ) )
12 simpl 444 . . . . . . 7  |-  ( ( A  e.  V  /\  y  =  B )  ->  A  e.  V )
13 abfmpel.3 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1413ancoms 440 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( ph  <->  ps )
)
1514adantll 695 . . . . . . 7  |-  ( ( ( A  e.  V  /\  y  =  B
)  /\  x  =  A )  ->  ( ph 
<->  ps ) )
1612, 15sbcied 3142 . . . . . 6  |-  ( ( A  e.  V  /\  y  =  B )  ->  ( [. A  /  x ]. ph  <->  ps )
)
1716ex 424 . . . . 5  |-  ( A  e.  V  ->  (
y  =  B  -> 
( [. A  /  x ]. ph  <->  ps ) ) )
1817alrimiv 1638 . . . 4  |-  ( A  e.  V  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )
19 elabgt 3024 . . . 4  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )  ->  ( B  e.  { y  |  [. A  /  x ]. ph }  <->  ps )
)
2018, 19sylan2 461 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
2120ancoms 440 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
2211, 21bitrd 245 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   {cab 2375   _Vcvv 2901   [.wsbc 3106   [_csb 3196    e. cmpt 4209   ` cfv 5396
This theorem is referenced by:  issiga  24292  ismeas  24351
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404
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