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Theorem imainss 5178
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 2867 . . . . . . . . . . 11  |-  y  e. 
_V
2 vex 2867 . . . . . . . . . . 11  |-  x  e. 
_V
31, 2brcnv 4946 . . . . . . . . . 10  |-  ( y `' R x  <->  x R
y )
4 19.8a 1747 . . . . . . . . . 10  |-  ( ( y  e.  B  /\  y `' R x )  ->  E. y ( y  e.  B  /\  y `' R x ) )
53, 4sylan2br 462 . . . . . . . . 9  |-  ( ( y  e.  B  /\  x R y )  ->  E. y ( y  e.  B  /\  y `' R x ) )
65ancoms 439 . . . . . . . 8  |-  ( ( x R y  /\  y  e.  B )  ->  E. y ( y  e.  B  /\  y `' R x ) )
76anim2i 552 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  (
x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
8 simprl 732 . . . . . . 7  |-  ( ( x  e.  A  /\  ( x R y  /\  y  e.  B
) )  ->  x R y )
97, 8jca 518 . . . . . 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 629 . . . . 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 3434 . . . . . . 7  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " B ) ) )
122elima2 5100 . . . . . . . 8  |-  ( x  e.  ( `' R " B )  <->  E. y
( y  e.  B  /\  y `' R x ) )
1312anbi2i 675 . . . . . . 7  |-  ( ( x  e.  A  /\  x  e.  ( `' R " B ) )  <-> 
( x  e.  A  /\  E. y ( y  e.  B  /\  y `' R x ) ) )
1411, 13bitri 240 . . . . . 6  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  E. y
( y  e.  B  /\  y `' R x ) ) )
1514anbi1i 676 . . . . 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 203 . . . 4  |-  ( ( ( x  e.  A  /\  x R y )  /\  y  e.  B
)  ->  ( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
1716eximi 1576 . . 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 5100 . . . . 5  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
1918anbi1i 676 . . . 4  |-  ( ( y  e.  ( R
" A )  /\  y  e.  B )  <->  ( E. x ( x  e.  A  /\  x R y )  /\  y  e.  B )
)
20 elin 3434 . . . 4  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  ( y  e.  ( R " A
)  /\  y  e.  B ) )
21 19.41v 1906 . . . 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 268 . . 3  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  E. x
( ( x  e.  A  /\  x R y )  /\  y  e.  B ) )
231elima2 5100 . . 3  |-  ( y  e.  ( R "
( A  i^i  ( `' R " B ) ) )  <->  E. x
( x  e.  ( A  i^i  ( `' R " B ) )  /\  x R y ) )
2417, 22, 233imtr4i 257 . 2  |-  ( y  e.  ( ( R
" A )  i^i 
B )  ->  y  e.  ( R " ( A  i^i  ( `' R " B ) ) ) )
2524ssriv 3260 1  |-  ( ( R " A )  i^i  B )  C_  ( R " ( A  i^i  ( `' R " B ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1541    e. wcel 1710    i^i cin 3227    C_ wss 3228   class class class wbr 4104   `'ccnv 4770   "cima 4774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784
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