MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imainss Unicode version

Theorem imainss 5096
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 2791 . . . . . . . . . . 11  |-  y  e. 
_V
2 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
31, 2brcnv 4864 . . . . . . . . . 10  |-  ( y `' R x  <->  x R
y )
4 19.8a 1718 . . . . . . . . . 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 3358 . . . . . . 7  |-  ( x  e.  ( A  i^i  ( `' R " B ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " B ) ) )
122elima2 5018 . . . . . . . 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 1563 . . 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 5018 . . . . 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 3358 . . . 4  |-  ( y  e.  ( ( R
" A )  i^i 
B )  <->  ( y  e.  ( R " A
)  /\  y  e.  B ) )
21 19.41v 1842 . . . 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 5018 . . 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 3184 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 1528    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   "cima 4692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
  Copyright terms: Public domain W3C validator