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Theorem imainlem 5108
Description: One direction of imain 5109. This direction does not require  Fun  `' F. (Contributed by Jim Kingdon, 25-Dec-2018.)
Assertion
Ref Expression
imainlem  |-  ( F
" ( A  i^i  B ) )  C_  (
( F " A
)  i^i  ( F " B ) )

Proof of Theorem imainlem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2366 . . . . 5  |-  ( E. x  e.  ( A  i^i  B ) x F y  <->  E. x
( x  e.  ( A  i^i  B )  /\  x F y ) )
2 elin 3184 . . . . . . . . 9  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
32anbi1i 447 . . . . . . . 8  |-  ( ( x  e.  ( A  i^i  B )  /\  x F y )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  x F y ) )
4 anandir 559 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x F
y )  <->  ( (
x  e.  A  /\  x F y )  /\  ( x  e.  B  /\  x F y ) ) )
53, 4bitri 183 . . . . . . 7  |-  ( ( x  e.  ( A  i^i  B )  /\  x F y )  <->  ( (
x  e.  A  /\  x F y )  /\  ( x  e.  B  /\  x F y ) ) )
65exbii 1542 . . . . . 6  |-  ( E. x ( x  e.  ( A  i^i  B
)  /\  x F
y )  <->  E. x
( ( x  e.  A  /\  x F y )  /\  (
x  e.  B  /\  x F y ) ) )
7 19.40 1568 . . . . . 6  |-  ( E. x ( ( x  e.  A  /\  x F y )  /\  ( x  e.  B  /\  x F y ) )  ->  ( E. x ( x  e.  A  /\  x F y )  /\  E. x ( x  e.  B  /\  x F y ) ) )
86, 7sylbi 120 . . . . 5  |-  ( E. x ( x  e.  ( A  i^i  B
)  /\  x F
y )  ->  ( E. x ( x  e.  A  /\  x F y )  /\  E. x ( x  e.  B  /\  x F y ) ) )
91, 8sylbi 120 . . . 4  |-  ( E. x  e.  ( A  i^i  B ) x F y  ->  ( E. x ( x  e.  A  /\  x F y )  /\  E. x ( x  e.  B  /\  x F y ) ) )
10 df-rex 2366 . . . . 5  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
11 df-rex 2366 . . . . 5  |-  ( E. x  e.  B  x F y  <->  E. x
( x  e.  B  /\  x F y ) )
1210, 11anbi12i 449 . . . 4  |-  ( ( E. x  e.  A  x F y  /\  E. x  e.  B  x F y )  <->  ( E. x ( x  e.  A  /\  x F y )  /\  E. x ( x  e.  B  /\  x F y ) ) )
139, 12sylibr 133 . . 3  |-  ( E. x  e.  ( A  i^i  B ) x F y  ->  ( E. x  e.  A  x F y  /\  E. x  e.  B  x F y ) )
1413ss2abi 3094 . 2  |-  { y  |  E. x  e.  ( A  i^i  B
) x F y }  C_  { y  |  ( E. x  e.  A  x F
y  /\  E. x  e.  B  x F
y ) }
15 dfima2 4789 . 2  |-  ( F
" ( A  i^i  B ) )  =  {
y  |  E. x  e.  ( A  i^i  B
) x F y }
16 dfima2 4789 . . . 4  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
17 dfima2 4789 . . . 4  |-  ( F
" B )  =  { y  |  E. x  e.  B  x F y }
1816, 17ineq12i 3200 . . 3  |-  ( ( F " A )  i^i  ( F " B ) )  =  ( { y  |  E. x  e.  A  x F y }  i^i  { y  |  E. x  e.  B  x F
y } )
19 inab 3268 . . 3  |-  ( { y  |  E. x  e.  A  x F
y }  i^i  {
y  |  E. x  e.  B  x F
y } )  =  { y  |  ( E. x  e.  A  x F y  /\  E. x  e.  B  x F y ) }
2018, 19eqtri 2109 . 2  |-  ( ( F " A )  i^i  ( F " B ) )  =  { y  |  ( E. x  e.  A  x F y  /\  E. x  e.  B  x F y ) }
2114, 15, 203sstr4i 3066 1  |-  ( F
" ( A  i^i  B ) )  C_  (
( F " A
)  i^i  ( F " B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103   E.wex 1427    e. wcel 1439   {cab 2075   E.wrex 2361    i^i cin 2999    C_ wss 3000   class class class wbr 3851   "cima 4454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-cnv 4459  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464
This theorem is referenced by:  imain  5109
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