Theorem imainlem 5000

Description: One direction of imain 5001. 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 2354	. . . . 5 |- ( E. x e. ( A i^i B ) x F y <-> E. x ( x e. ( A i^i B ) /\ x F y ) )
2		elin 3155	. . . . . . . . 9 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
3	2	anbi1i 445	. . . . . . . 8 |- ( ( x e. ( A i^i B ) /\ x F y ) <-> ( ( x e. A /\ x e. B ) /\ x F y ) )
4		anandir 555	. . . . . . . 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 182	. . . . . . 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 1536	. . . . . 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 1562	. . . . . 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 119	. . . . 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 119	. . . 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 2354	. . . . 5 |- ( E. x e. A x F y <-> E. x ( x e. A /\ x F y ) )
11		df-rex 2354	. . . . 5 |- ( E. x e. B x F y <-> E. x ( x e. B /\ x F y ) )
12	10, 11	anbi12i 447	. . . 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 132	. . 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 3066	. 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 4690	. 2 |- ( F " ( A i^i B ) ) = { y | E. x e. ( A i^i B ) x F y }
16		dfima2 4690	. . . 4 |- ( F " A ) = { y | E. x e. A x F y }
17		dfima2 4690	. . . 4 |- ( F " B ) = { y | E. x e. B x F y }
18	16, 17	ineq12i 3165	. . 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 3232	. . 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 2101	. 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 3038	1 |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) )

Syntax hints:   /\ wa 102   E.wex 1421   e. wcel 1433   {cab 2067   E.wrex 2349   i^i cin 2972   C_ wss 2973   class class class wbr 3785   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  imain  5001