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Theorem abfmpeld 24068
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 1642 . . . . . . . . 9  |-  ( ph  ->  A. x { y  |  ps }  e.  _V )
3 csbexg 3263 . . . . . . . . 9  |-  ( ( A  e.  V  /\  A. x { y  |  ps }  e.  _V )  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
42, 3sylan2 462 . . . . . . . 8  |-  ( ( A  e.  V  /\  ph )  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
5 abfmpeld.1 . . . . . . . . 9  |-  F  =  ( x  e.  V  |->  { y  |  ps } )
65fvmpts 5809 . . . . . . . 8  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ps }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ps } )
74, 6syldan 458 . . . . . . 7  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [_ A  /  x ]_ {
y  |  ps }
)
8 csbabg 3312 . . . . . . . . 9  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ps }  =  { y  |  [. A  /  x ]. ps } )
98eqeq2d 2449 . . . . . . . 8  |-  ( A  e.  V  ->  (
( F `  A
)  =  [_ A  /  x ]_ { y  |  ps }  <->  ( F `  A )  =  {
y  |  [. A  /  x ]. ps }
) )
109adantr 453 . . . . . . 7  |-  ( ( A  e.  V  /\  ph )  ->  ( ( F `  A )  =  [_ A  /  x ]_ { y  |  ps } 
<->  ( F `  A
)  =  { y  |  [. A  /  x ]. ps } ) )
117, 10mpbid 203 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  {
y  |  [. A  /  x ]. ps }
)
1211eleq2d 2505 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  ( B  e.  ( F `  A
)  <->  B  e.  { y  |  [. A  /  x ]. ps } ) )
1312adantl 454 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ps } ) )
14 simpll 732 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  A  e.  V )
15 abfmpeld.3 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
1615ancomsd 442 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  =  B  /\  x  =  A )  ->  ( ps 
<->  ch ) ) )
1716adantl 454 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ph )  ->  ( (
y  =  B  /\  x  =  A )  ->  ( ps  <->  ch )
) )
1817impl 605 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  ph )  /\  y  =  B
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1914, 18sbcied 3199 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  ( [. A  /  x ]. ps  <->  ch ) )
2019ex 425 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )
2120alrimiv 1642 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch )
) )
22 elabgt 3081 . . . . 5  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2321, 22sylan2 462 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2413, 23bitrd 246 . . 3  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2524an13s 780 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2625ex 425 1  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958   [.wsbc 3163   [_csb 3253    e. cmpt 4268   ` cfv 5456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464
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