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Theorem imainss 5270
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 3083 . . . . . . . . . . 11  |-  y  e. 
_V
2 vex 3083 . . . . . . . . . . 11  |-  x  e. 
_V
31, 2brcnv 5036 . . . . . . . . . 10  |-  ( y `' R x  <->  x R
y )
4 19.8a 1912 . . . . . . . . . 10  |-  ( ( y  e.  B  /\  y `' R x )  ->  E. y ( y  e.  B  /\  y `' R x ) )
53, 4sylan2br 478 . . . . . . . . 9  |-  ( ( y  e.  B  /\  x R y )  ->  E. y ( y  e.  B  /\  y `' R x ) )
65ancoms 454 . . . . . . . 8  |-  ( ( x R y  /\  y  e.  B )  ->  E. y ( y  e.  B  /\  y `' R x ) )
76anim2i 571 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  (
x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
8 simprl 762 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  x R y )
97, 8jca 534 . . . . . 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 652 . . . . 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 3649 . . . . . . 7  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " B ) ) )
122elima2 5193 . . . . . . . 8  |-  ( x  e.  ( `' R " B )  <->  E. y
( y  e.  B  /\  y `' R x ) )
1312anbi2i 698 . . . . . . 7  |-  ( ( x  e.  A  /\  x  e.  ( `' R " B ) )  <-> 
( x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
1411, 13bitri 252 . . . . . 6  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  y `' R x ) ) )
1514anbi1i 699 . . . . 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 215 . . . 4  |-  ( ( ( x  e.  A  /\  x R y )  /\  y  e.  B
)  ->  ( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
1716eximi 1701 . . 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 5193 . . . . 5  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
1918anbi1i 699 . . . 4  |-  ( ( y  e.  ( R
" A )  /\  y  e.  B )  <->  ( E. x ( x  e.  A  /\  x R y )  /\  y  e.  B )
)
20 elin 3649 . . . 4  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  ( y  e.  ( R " A
)  /\  y  e.  B ) )
21 19.41v 1823 . . . 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 280 . . 3  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  E. x
( ( x  e.  A  /\  x R y )  /\  y  e.  B ) )
231elima2 5193 . . 3  |-  ( y  e.  ( R "
( A  i^i  ( `' R " B ) ) )  <->  E. x
( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
2417, 22, 233imtr4i 269 . 2  |-  ( y  e.  ( ( R
" A )  i^i 
B )  ->  y  e.  ( R " ( A  i^i  ( `' R " B ) ) ) )
2524ssriv 3468 1  |-  ( ( R " A )  i^i  B )  C_  ( R " ( A  i^i  ( `' R " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370   E.wex 1657    e. wcel 1872    i^i cin 3435    C_ wss 3436   class class class wbr 4423   `'ccnv 4852   "cima 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-cnv 4861  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866
This theorem is referenced by: (None)
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