Theorem imainlem 5418

Description: One direction of imain 5419. 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 2517	. . . . 5 |- ( E. x e. ( A i^i B ) x F y <-> E. x ( x e. ( A i^i B ) /\ x F y ) )
2		elin 3392	. . . . . . . . 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 595	. . . . . . . 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 1654	. . . . . 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 1680	. . . . . 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 2517	. . . . 5 |- ( E. x e. A x F y <-> E. x ( x e. A /\ x F y ) )
11		df-rex 2517	. . . . 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 3300	. 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 5084	. 2 |- ( F " ( A i^i B ) ) = { y | E. x e. ( A i^i B ) x F y }
16		dfima2 5084	. . . 4 |- ( F " A ) = { y | E. x e. A x F y }
17		dfima2 5084	. . . 4 |- ( F " B ) = { y | E. x e. B x F y }
18	16, 17	ineq12i 3408	. . 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 3477	. . 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 2252	. 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 3269	1 |- ( F " ( A i^i B ) ) C_ ( ( F " A ) i^i ( F " B ) )

Syntax hints:   /\ wa 104   E.wex 1541   e. wcel 2202   {cab 2217   E.wrex 2512   i^i cin 3200   C_ wss 3201   class class class wbr 4093   "cima 4734

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  imain  5419