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Theorem abfmpel 27153
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 4575 . . . . . 6  |-  [_ A  /  x ]_ { y  |  ph }  e.  _V
3 abfmpel.1 . . . . . . 7  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
43fvmpts 5945 . . . . . 6  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ph }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ph } )
52, 4mpan2 671 . . . . 5  |-  ( A  e.  V  ->  ( F `  A )  =  [_ A  /  x ]_ { y  |  ph } )
6 csbab 3850 . . . . 5  |-  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
75, 6syl6eq 2519 . . . 4  |-  ( A  e.  V  ->  ( F `  A )  =  { y  |  [. A  /  x ]. ph }
)
87eleq2d 2532 . . 3  |-  ( A  e.  V  ->  ( B  e.  ( F `  A )  <->  B  e.  { y  |  [. A  /  x ]. ph }
) )
98adantr 465 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ph } ) )
10 simpl 457 . . . . . . 7  |-  ( ( A  e.  V  /\  y  =  B )  ->  A  e.  V )
11 abfmpel.3 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1211ancoms 453 . . . . . . . 8  |-  ( ( y  =  B  /\  x  =  A )  ->  ( ph  <->  ps )
)
1312adantll 713 . . . . . . 7  |-  ( ( ( A  e.  V  /\  y  =  B
)  /\  x  =  A )  ->  ( ph 
<->  ps ) )
1410, 13sbcied 3363 . . . . . 6  |-  ( ( A  e.  V  /\  y  =  B )  ->  ( [. A  /  x ]. ph  <->  ps )
)
1514ex 434 . . . . 5  |-  ( A  e.  V  ->  (
y  =  B  -> 
( [. A  /  x ]. ph  <->  ps ) ) )
1615alrimiv 1690 . . . 4  |-  ( A  e.  V  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )
17 elabgt 3242 . . . 4  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )  ->  ( B  e.  { y  |  [. A  /  x ]. ph }  <->  ps )
)
1816, 17sylan2 474 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
1918ancoms 453 . 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 369   A.wal 1372    = wceq 1374    e. wcel 1762   {cab 2447   _Vcvv 3108   [.wsbc 3326   [_csb 3430    |-> cmpt 4500   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589
This theorem is referenced by:  issiga  27739  ismeas  27798
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