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Theorem abfmpel 23234
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 . . . . . . . 8  |-  { y  |  ph }  e.  _V
21ax-gen 1536 . . . . . . 7  |-  A. x { y  |  ph }  e.  _V
3 csbexg 3104 . . . . . . 7  |-  ( ( A  e.  V  /\  A. x { y  | 
ph }  e.  _V )  ->  [_ A  /  x ]_ { y  |  ph }  e.  _V )
42, 3mpan2 652 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  e.  _V )
5 abfmpel.1 . . . . . . 7  |-  F  =  ( x  e.  V  |->  { y  |  ph } )
65fvmpts 5619 . . . . . 6  |-  ( ( A  e.  V  /\  [_ A  /  x ]_ { y  |  ph }  e.  _V )  ->  ( F `  A
)  =  [_ A  /  x ]_ { y  |  ph } )
74, 6mpdan 649 . . . . 5  |-  ( A  e.  V  ->  ( F `  A )  =  [_ A  /  x ]_ { y  |  ph } )
8 csbabg 3155 . . . . . 6  |-  ( A  e.  V  ->  [_ A  /  x ]_ { y  |  ph }  =  { y  |  [. A  /  x ]. ph }
)
98eqeq2d 2307 . . . . 5  |-  ( A  e.  V  ->  (
( F `  A
)  =  [_ A  /  x ]_ { y  |  ph }  <->  ( F `  A )  =  {
y  |  [. A  /  x ]. ph }
) )
107, 9mpbid 201 . . . 4  |-  ( A  e.  V  ->  ( F `  A )  =  { y  |  [. A  /  x ]. ph }
)
11 eleq2 2357 . . . 4  |-  ( ( F `  A )  =  { y  | 
[. A  /  x ]. ph }  ->  ( B  e.  ( F `  A )  <->  B  e.  { y  |  [. A  /  x ]. ph }
) )
1210, 11syl 15 . . 3  |-  ( A  e.  V  ->  ( B  e.  ( F `  A )  <->  B  e.  { y  |  [. A  /  x ]. ph }
) )
1312adantr 451 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <-> 
B  e.  { y  |  [. A  /  x ]. ph } ) )
14 simpl 443 . . . . . . 7  |-  ( ( A  e.  V  /\  y  =  B )  ->  A  e.  V )
15 abfmpel.3 . . . . . . . . . 10  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1615ancoms 439 . . . . . . . . 9  |-  ( ( y  =  B  /\  x  =  A )  ->  ( ph  <->  ps )
)
17163adant1 973 . . . . . . . 8  |-  ( ( A  e.  V  /\  y  =  B  /\  x  =  A )  ->  ( ph  <->  ps )
)
18173expa 1151 . . . . . . 7  |-  ( ( ( A  e.  V  /\  y  =  B
)  /\  x  =  A )  ->  ( ph 
<->  ps ) )
1914, 18sbcied 3040 . . . . . 6  |-  ( ( A  e.  V  /\  y  =  B )  ->  ( [. A  /  x ]. ph  <->  ps )
)
2019ex 423 . . . . 5  |-  ( A  e.  V  ->  (
y  =  B  -> 
( [. A  /  x ]. ph  <->  ps ) ) )
2120alrimiv 1621 . . . 4  |-  ( A  e.  V  ->  A. y
( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )
22 elabgt 2924 . . . 4  |-  ( ( B  e.  W  /\  A. y ( y  =  B  ->  ( [. A  /  x ]. ph  <->  ps )
) )  ->  ( B  e.  { y  |  [. A  /  x ]. ph }  <->  ps )
)
2321, 22sylan2 460 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
2423ancoms 439 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  {
y  |  [. A  /  x ]. ph }  <->  ps ) )
2513, 24bitrd 244 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   [.wsbc 3004   [_csb 3094    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  issiga  23487  ismeas  23545  isibfm  23608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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