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Theorem abfmpel 27922
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 . . . . . . 7  |-  { y  |  ph }  e.  _V
21csbex 4528 . . . . . 6  |-  [_ A  /  x ]_ { y  |  ph }  e.  _V
3 abfmpel.1 . . . . . . 7  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
43fvmpts 5934 . . . . . 6  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ph }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ph } )
52, 4mpan2 669 . . . . 5  |-  ( A  e.  V  ->  ( F `  A )  =  [_ A  /  x ]_ { y  |  ph } )
6 csbab 3801 . . . . 5  |-  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
75, 6syl6eq 2459 . . . 4  |-  ( A  e.  V  ->  ( F `  A )  =  { y  |  [. A  /  x ]. ph }
)
87eleq2d 2472 . . 3  |-  ( A  e.  V  ->  ( B  e.  ( F `  A )  <->  B  e.  { y  |  [. A  /  x ]. ph }
) )
98adantr 463 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ph } ) )
10 simpl 455 . . . . . . 7  |-  ( ( A  e.  V  /\  y  =  B )  ->  A  e.  V )
11 abfmpel.3 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1211ancoms 451 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( ph  <->  ps )
)
1312adantll 712 . . . . . . 7  |-  ( ( ( A  e.  V  /\  y  =  B
)  /\  x  =  A )  ->  ( ph 
<->  ps ) )
1410, 13sbcied 3313 . . . . . 6  |-  ( ( A  e.  V  /\  y  =  B )  ->  ( [. A  /  x ]. ph  <->  ps )
)
1514ex 432 . . . . 5  |-  ( A  e.  V  ->  (
y  =  B  -> 
( [. A  /  x ]. ph  <->  ps ) ) )
1615alrimiv 1740 . . . 4  |-  ( A  e.  V  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )
17 elabgt 3192 . . . 4  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )  ->  ( B  e.  { y  |  [. A  /  x ]. ph }  <->  ps )
)
1816, 17sylan2 472 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
1918ancoms 451 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
209, 19bitrd 253 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3058   [.wsbc 3276   [_csb 3372    |-> cmpt 4452   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576
This theorem is referenced by:  issiga  28545  ismeas  28633
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