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Theorem abfmpeld 27632
Description: Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
Hypotheses
Ref Expression
abfmpeld.1  |-  F  =  ( x  e.  V  |->  { y  |  ps } )
abfmpeld.2  |-  ( ph  ->  { y  |  ps }  e.  _V )
abfmpeld.3  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
Assertion
Ref Expression
abfmpeld  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch )
) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    x, V, y   
y, W    ch, x, y    ph, x, y
Allowed substitution hints:    ps( x, y)    W( x)

Proof of Theorem abfmpeld
StepHypRef Expression
1 abfmpeld.2 . . . . . . . . . 10  |-  ( ph  ->  { y  |  ps }  e.  _V )
21alrimiv 1727 . . . . . . . . 9  |-  ( ph  ->  A. x { y  |  ps }  e.  _V )
3 csbexg 4499 . . . . . . . . 9  |-  ( A. x { y  |  ps }  e.  _V  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
42, 3syl 16 . . . . . . . 8  |-  ( ph  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
5 abfmpeld.1 . . . . . . . . 9  |-  F  =  ( x  e.  V  |->  { y  |  ps } )
65fvmpts 5859 . . . . . . . 8  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ps }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ps } )
74, 6sylan2 472 . . . . . . 7  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [_ A  /  x ]_ {
y  |  ps }
)
8 csbab 3775 . . . . . . 7  |-  [_ A  /  x ]_ { y  |  ps }  =  { y  |  [. A  /  x ]. ps }
97, 8syl6eq 2439 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  {
y  |  [. A  /  x ]. ps }
)
109eleq2d 2452 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  ( B  e.  ( F `  A
)  <->  B  e.  { y  |  [. A  /  x ]. ps } ) )
1110adantl 464 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ps } ) )
12 simpll 751 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  A  e.  V )
13 abfmpeld.3 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
1413ancomsd 452 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  =  B  /\  x  =  A )  ->  ( ps 
<->  ch ) ) )
1514adantl 464 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ph )  ->  ( (
y  =  B  /\  x  =  A )  ->  ( ps  <->  ch )
) )
1615impl 618 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  ph )  /\  y  =  B
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1712, 16sbcied 3289 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  ( [. A  /  x ]. ps  <->  ch ) )
1817ex 432 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )
1918alrimiv 1727 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch )
) )
20 elabgt 3168 . . . . 5  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2119, 20sylan2 472 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2211, 21bitrd 253 . . 3  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2322an13s 801 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2423ex 432 1  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397    = wceq 1399    e. wcel 1826   {cab 2367   _Vcvv 3034   [.wsbc 3252   [_csb 3348    |-> cmpt 4425   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504
This theorem is referenced by: (None)
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