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Theorem imainlem 5174
Description: One direction of imain 5175. 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 2399 . . . . 5  |-  ( E. x  e.  ( A  i^i  B ) x F y  <->  E. x
( x  e.  ( A  i^i  B )  /\  x F y ) )
2 elin 3229 . . . . . . . . 9  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
32anbi1i 453 . . . . . . . 8  |-  ( ( x  e.  ( A  i^i  B )  /\  x F y )  <->  ( (
x  e.  A  /\  x  e.  B )  /\  x F y ) )
4 anandir 565 . . . . . . . 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 1569 . . . . . 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 1595 . . . . . 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 2399 . . . . 5  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
11 df-rex 2399 . . . . 5  |-  ( E. x  e.  B  x F y  <->  E. x
( x  e.  B  /\  x F y ) )
1210, 11anbi12i 455 . . . 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 3139 . 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 4853 . 2  |-  ( F
" ( A  i^i  B ) )  =  {
y  |  E. x  e.  ( A  i^i  B
) x F y }
16 dfima2 4853 . . . 4  |-  ( F
" A )  =  { y  |  E. x  e.  A  x F y }
17 dfima2 4853 . . . 4  |-  ( F
" B )  =  { y  |  E. x  e.  B  x F y }
1816, 17ineq12i 3245 . . 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 3314 . . 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 2138 . 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 3108 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 1453    e. wcel 1465   {cab 2103   E.wrex 2394    i^i cin 3040    C_ wss 3041   class class class wbr 3899   "cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  imain  5175
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