Theorem imainlem 5339

Description: One direction of imain 5340. 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 2481	. . . . 5 |- ( E. x e. ( A i^i B ) x F y <-> E. x ( x e. ( A i^i B ) /\ x F y ) )
2		elin 3346	. . . . . . . . 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 1619	. . . . . 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 1645	. . . . . 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 2481	. . . . 5 |- ( E. x e. A x F y <-> E. x ( x e. A /\ x F y ) )
11		df-rex 2481	. . . . 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 3255	. 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 5011	. 2 |- ( F " ( A i^i B ) ) = { y | E. x e. ( A i^i B ) x F y }
16		dfima2 5011	. . . 4 |- ( F " A ) = { y | E. x e. A x F y }
17		dfima2 5011	. . . 4 |- ( F " B ) = { y | E. x e. B x F y }
18	16, 17	ineq12i 3362	. . 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 3431	. . 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 2217	. 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 3224	1 |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) )

Syntax hints:   /\ wa 104   E.wex 1506   e. wcel 2167   {cab 2182   E.wrex 2476   i^i cin 3156   C_ wss 3157   class class class wbr 4033   "cima 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  imain  5340