Theorem imainlem 5355

Description: One direction of imain 5356. 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.

Step	Hyp	Ref	Expression
1		df-rex 2490	. . . . 5 |- ( E. x e. ( A i^i B ) x F y <-> E. x ( x e. ( A i^i B ) /\ x F y ) )
2		elin 3356	. . . . . . . . 9 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3	2	anbi1i 458	. . . . . . . 8 |- ( ( x e. ( A i^i B ) /\ x F y ) <-> ( ( x e. A /\ x e. B ) /\ x F y ) )
4		anandir 591	. . . . . . . 8 |- ( ( ( x e. A /\ x e. B ) /\ x F y ) <-> ( ( x e. A /\ x F y ) /\ ( x e. B /\ x F y ) ) )
5	3, 4	bitri 184	. . . . . . 7 |- ( ( x e. ( A i^i B ) /\ x F y ) <-> ( ( x e. A /\ x F y ) /\ ( x e. B /\ x F y ) ) )
6	5	exbii 1628	. . . . . 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 1654	. . . . . 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 ) ) )
8	6, 7	sylbi 121	. . . . 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 ) ) )
9	1, 8	sylbi 121	. . . 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 2490	. . . . 5 |- ( E. x e. A x F y <-> E. x ( x e. A /\ x F y ) )
11		df-rex 2490	. . . . 5 |- ( E. x e. B x F y <-> E. x ( x e. B /\ x F y ) )
12	10, 11	anbi12i 460	. . . 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 ) ) )
13	9, 12	sylibr 134	. . 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 ) )
14	13	ss2abi 3265	. 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 5024	. 2 |- ( F " ( A i^i B ) ) = { y | E. x e. ( A i^i B ) x F y }
16		dfima2 5024	. . . 4 |- ( F " A ) = { y | E. x e. A x F y }
17		dfima2 5024	. . . 4 |- ( F " B ) = { y | E. x e. B x F y }
18	16, 17	ineq12i 3372	. . 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 3441	. . 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 ) }
20	18, 19	eqtri 2226	. 2 |- ( ( F " A ) i^i ( F " B ) ) = { y | ( E. x e. A x F y /\ E. x e. B x F y ) }
21	14, 15, 20	3sstr4i 3234	1 |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) )

Syntax hints:   /\ wa 104   E.wex 1515   e. wcel 2176   {cab 2191   E.wrex 2485   i^i cin 3165   C_ wss 3166   class class class wbr 4044   "cima 4678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by:  imain  5356