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Theorem abfmpeld 28253
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 1764 . . . . . . . . 9  |-  ( ph  ->  A. x { y  |  ps }  e.  _V )
3 csbexg 4557 . . . . . . . . 9  |-  ( A. x { y  |  ps }  e.  _V  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
42, 3syl 17 . . . . . . . 8  |-  ( ph  ->  [_ A  /  x ]_ { y  |  ps }  e.  _V )
5 abfmpeld.1 . . . . . . . . 9  |-  F  =  ( x  e.  V  |->  { y  |  ps } )
65fvmpts 5966 . . . . . . . 8  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ps }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ps } )
74, 6sylan2 477 . . . . . . 7  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [_ A  /  x ]_ {
y  |  ps }
)
8 csbab 3827 . . . . . . 7  |-  [_ A  /  x ]_ { y  |  ps }  =  { y  |  [. A  /  x ]. ps }
97, 8syl6eq 2480 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  {
y  |  [. A  /  x ]. ps }
)
109eleq2d 2493 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  ( B  e.  ( F `  A
)  <->  B  e.  { y  |  [. A  /  x ]. ps } ) )
1110adantl 468 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ps } ) )
12 simpll 759 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  A  e.  V )
13 abfmpeld.3 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
1413ancomsd 456 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  =  B  /\  x  =  A )  ->  ( ps 
<->  ch ) ) )
1514adantl 468 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ph )  ->  ( (
y  =  B  /\  x  =  A )  ->  ( ps  <->  ch )
) )
1615impl 625 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  ph )  /\  y  =  B
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1712, 16sbcied 3338 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  ( [. A  /  x ]. ps  <->  ch ) )
1817ex 436 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )
1918alrimiv 1764 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch )
) )
20 elabgt 3216 . . . . 5  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2119, 20sylan2 477 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2211, 21bitrd 257 . . 3  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2322an13s 811 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  W ) )  -> 
( B  e.  ( F `  A )  <->  ch ) )
2423ex 436 1  |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1436    = wceq 1438    e. wcel 1869   {cab 2408   _Vcvv 3082   [.wsbc 3301   [_csb 3397    |-> cmpt 4481   ` cfv 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4545  ax-nul 4554  ax-pr 4659
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608
This theorem is referenced by: (None)
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