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Theorem imainlem 5008
Description: One direction of imain 5009. 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 2329 . . . . 5  |-  ( E. x  e.  ( A  i^i  B ) x F y  <->  E. x
( x  e.  ( A  i^i  B )  /\  x F y ) )
2 elin 3154 . . . . . . . . 9  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
32anbi1i 439 . . . . . . . 8  |-  ( ( x  e.  ( A  i^i  B )  /\  x F y )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  x F y ) )
4 anandir 533 . . . . . . . 8  |-  ( ( ( x  e.  A  /\  x  e.  B
)  /\  x F
y )  <->  ( (
x  e.  A  /\  x F y )  /\  ( x  e.  B  /\  x F y ) ) )
53, 4bitri 177 . . . . . . 7  |-  ( ( x  e.  ( A  i^i  B )  /\  x F y )  <->  ( (
x  e.  A  /\  x F y )  /\  ( x  e.  B  /\  x F y ) ) )
65exbii 1512 . . . . . 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 1538 . . . . . 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 118 . . . . 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 118 . . . 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 2329 . . . . 5  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
11 df-rex 2329 . . . . 5  |-  ( E. x  e.  B  x F y  <->  E. x
( x  e.  B  /\  x F y ) )
1210, 11anbi12i 441 . . . 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 141 . . 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 3040 . 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 4698 . 2  |-  ( F
" ( A  i^i  B ) )  =  {
y  |  E. x  e.  ( A  i^i  B
) x F y }
16 dfima2 4698 . . . 4  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
17 dfima2 4698 . . . 4  |-  ( F
" B )  =  { y  |  E. x  e.  B  x F y }
1816, 17ineq12i 3164 . . 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 3233 . . 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 2076 . 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 3012 1  |-  ( F
" ( A  i^i  B ) )  C_  (
( F " A
)  i^i  ( F " B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324    i^i cin 2944    C_ wss 2945   class class class wbr 3792   "cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by:  imain  5009
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