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Theorem abfmpeld 27164
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 1695 . . . . . . . . 9  |-  ( ph  ->  A. x { y  |  ps }  e.  _V )
3 csbexg 4579 . . . . . . . . 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 5950 . . . . . . . 8  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ps }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ps } )
74, 6sylan2 474 . . . . . . 7  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [_ A  /  x ]_ {
y  |  ps }
)
8 csbab 3855 . . . . . . 7  |-  [_ A  /  x ]_ { y  |  ps }  =  { y  |  [. A  /  x ]. ps }
97, 8syl6eq 2524 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  {
y  |  [. A  /  x ]. ps }
)
109eleq2d 2537 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  ( B  e.  ( F `  A
)  <->  B  e.  { y  |  [. A  /  x ]. ps } ) )
1110adantl 466 . . . 4  |-  ( ( B  e.  W  /\  ( A  e.  V  /\  ph ) )  -> 
( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ps } ) )
12 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  A  e.  V )
13 abfmpeld.3 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps 
<->  ch ) ) )
1413ancomsd 454 . . . . . . . . . 10  |-  ( ph  ->  ( ( y  =  B  /\  x  =  A )  ->  ( ps 
<->  ch ) ) )
1514adantl 466 . . . . . . . . 9  |-  ( ( A  e.  V  /\  ph )  ->  ( (
y  =  B  /\  x  =  A )  ->  ( ps  <->  ch )
) )
1615impl 620 . . . . . . . 8  |-  ( ( ( ( A  e.  V  /\  ph )  /\  y  =  B
)  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1712, 16sbcied 3368 . . . . . . 7  |-  ( ( ( A  e.  V  /\  ph )  /\  y  =  B )  ->  ( [. A  /  x ]. ps  <->  ch ) )
1817ex 434 . . . . . 6  |-  ( ( A  e.  V  /\  ph )  ->  ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )
1918alrimiv 1695 . . . . 5  |-  ( ( A  e.  V  /\  ph )  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch )
) )
20 elabgt 3247 . . . . 5  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ps  <->  ch ) ) )  -> 
( B  e.  {
y  |  [. A  /  x ]. ps }  <->  ch ) )
2119, 20sylan2 474 . . . 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 434 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 369   A.wal 1377    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113   [.wsbc 3331   [_csb 3435    |-> cmpt 4505   ` cfv 5586
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  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-iota 5549  df-fun 5588  df-fv 5594
This theorem is referenced by: (None)
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