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Theorem imainss 5227
Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
imainss  |-  ( ( R " A )  i^i  B )  C_  ( R " ( A  i^i  ( `' R " B ) ) )

Proof of Theorem imainss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2902 . . . . . . . . . . 11  |-  y  e. 
_V
2 vex 2902 . . . . . . . . . . 11  |-  x  e. 
_V
31, 2brcnv 4995 . . . . . . . . . 10  |-  ( y `' R x  <->  x R
y )
4 19.8a 1754 . . . . . . . . . 10  |-  ( ( y  e.  B  /\  y `' R x )  ->  E. y ( y  e.  B  /\  y `' R x ) )
53, 4sylan2br 463 . . . . . . . . 9  |-  ( ( y  e.  B  /\  x R y )  ->  E. y ( y  e.  B  /\  y `' R x ) )
65ancoms 440 . . . . . . . 8  |-  ( ( x R y  /\  y  e.  B )  ->  E. y ( y  e.  B  /\  y `' R x ) )
76anim2i 553 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  (
x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
8 simprl 733 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  x R y )
97, 8jca 519 . . . . . 6  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  (
( x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) )  /\  x R y ) )
109anassrs 630 . . . . 5  |-  ( ( ( x  e.  A  /\  x R y )  /\  y  e.  B
)  ->  ( (
x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) )  /\  x R y ) )
11 elin 3473 . . . . . . 7  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " B ) ) )
122elima2 5149 . . . . . . . 8  |-  ( x  e.  ( `' R " B )  <->  E. y
( y  e.  B  /\  y `' R x ) )
1312anbi2i 676 . . . . . . 7  |-  ( ( x  e.  A  /\  x  e.  ( `' R " B ) )  <-> 
( x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
1411, 13bitri 241 . . . . . 6  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  y `' R x ) ) )
1514anbi1i 677 . . . . 5  |-  ( ( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y )  <->  ( (
x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) )  /\  x R y ) )
1610, 15sylibr 204 . . . 4  |-  ( ( ( x  e.  A  /\  x R y )  /\  y  e.  B
)  ->  ( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
1716eximi 1582 . . 3  |-  ( E. x ( ( x  e.  A  /\  x R y )  /\  y  e.  B )  ->  E. x ( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
181elima2 5149 . . . . 5  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
1918anbi1i 677 . . . 4  |-  ( ( y  e.  ( R
" A )  /\  y  e.  B )  <->  ( E. x ( x  e.  A  /\  x R y )  /\  y  e.  B )
)
20 elin 3473 . . . 4  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  ( y  e.  ( R " A
)  /\  y  e.  B ) )
21 19.41v 1913 . . . 4  |-  ( E. x ( ( x  e.  A  /\  x R y )  /\  y  e.  B )  <->  ( E. x ( x  e.  A  /\  x R y )  /\  y  e.  B )
)
2219, 20, 213bitr4i 269 . . 3  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  E. x
( ( x  e.  A  /\  x R y )  /\  y  e.  B ) )
231elima2 5149 . . 3  |-  ( y  e.  ( R "
( A  i^i  ( `' R " B ) ) )  <->  E. x
( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
2417, 22, 233imtr4i 258 . 2  |-  ( y  e.  ( ( R
" A )  i^i 
B )  ->  y  e.  ( R " ( A  i^i  ( `' R " B ) ) ) )
2524ssriv 3295 1  |-  ( ( R " A )  i^i  B )  C_  ( R " ( A  i^i  ( `' R " B ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    e. wcel 1717    i^i cin 3262    C_ wss 3263   class class class wbr 4153   `'ccnv 4817   "cima 4821
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831
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