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Theorem elmapintab 36248
Description: Two ways to say a set is an element of mapped intersection of a class. Here  F maps elements of  C to elements of  |^| { x  | 
ph } or  x. (Contributed by RP, 19-Aug-2020.)
Hypotheses
Ref Expression
elmapintab.1  |-  ( A  e.  B  <->  ( A  e.  C  /\  ( F `  A )  e.  |^| { x  | 
ph } ) )
elmapintab.2  |-  ( A  e.  E  <->  ( A  e.  C  /\  ( F `  A )  e.  x ) )
Assertion
Ref Expression
elmapintab  |-  ( A  e.  B  <->  ( A  e.  C  /\  A. x
( ph  ->  A  e.  E ) ) )
Distinct variable groups:    x, A    x, C    x, F
Allowed substitution hints:    ph( x)    B( x)    E( x)

Proof of Theorem elmapintab
StepHypRef Expression
1 elmapintab.1 . 2  |-  ( A  e.  B  <->  ( A  e.  C  /\  ( F `  A )  e.  |^| { x  | 
ph } ) )
2 fvex 5902 . . . 4  |-  ( F `
 A )  e. 
_V
32elintab 4259 . . 3  |-  ( ( F `  A )  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  ( F `
 A )  e.  x ) )
43anbi2i 705 . 2  |-  ( ( A  e.  C  /\  ( F `  A )  e.  |^| { x  | 
ph } )  <->  ( A  e.  C  /\  A. x
( ph  ->  ( F `
 A )  e.  x ) ) )
5 elmapintab.2 . . . . . 6  |-  ( A  e.  E  <->  ( A  e.  C  /\  ( F `  A )  e.  x ) )
65baibr 920 . . . . 5  |-  ( A  e.  C  ->  (
( F `  A
)  e.  x  <->  A  e.  E ) )
76imbi2d 322 . . . 4  |-  ( A  e.  C  ->  (
( ph  ->  ( F `
 A )  e.  x )  <->  ( ph  ->  A  e.  E ) ) )
87albidv 1778 . . 3  |-  ( A  e.  C  ->  ( A. x ( ph  ->  ( F `  A )  e.  x )  <->  A. x
( ph  ->  A  e.  E ) ) )
98pm5.32i 647 . 2  |-  ( ( A  e.  C  /\  A. x ( ph  ->  ( F `  A )  e.  x ) )  <-> 
( A  e.  C  /\  A. x ( ph  ->  A  e.  E ) ) )
101, 4, 93bitri 279 1  |-  ( A  e.  B  <->  ( A  e.  C  /\  A. x
( ph  ->  A  e.  E ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453    e. wcel 1898   {cab 2448   |^|cint 4248   ` cfv 5605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-nul 4550
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-sn 3981  df-pr 3983  df-uni 4213  df-int 4249  df-iota 5569  df-fv 5613
This theorem is referenced by:  elcnvintab  36254
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