Theorem imainss 5144

Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
imainss	|- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Proof of Theorem imainss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2802	. . . . . . . . . . 11 |- y e. _V
2		vex 2802	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4905	. . . . . . . . . 10 |- ( y `' R x <-> x R y )
4		19.8a 1636	. . . . . . . . . 10 |- ( ( y e. B /\ y `' R x ) -> E. y ( y e. B /\ y `' R x ) )
5	3, 4	sylan2br 288	. . . . . . . . 9 |- ( ( y e. B /\ x R y ) -> E. y ( y e. B /\ y `' R x ) )
6	5	ancoms 268	. . . . . . . 8 |- ( ( x R y /\ y e. B ) -> E. y ( y e. B /\ y `' R x ) )
7	6	anim2i 342	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
8		simprl 529	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> x R y )
9	7, 8	jca 306	. . . . . 6 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
10	9	anassrs 400	. . . . 5 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
11		elin 3387	. . . . . . 7 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ x e. ( `' R " B ) ) )
12	2	elima2 5074	. . . . . . . 8 |- ( x e. ( `' R " B ) <-> E. y ( y e. B /\ y `' R x ) )
13	12	anbi2i 457	. . . . . . 7 |- ( ( x e. A /\ x e. ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
14	11, 13	bitri 184	. . . . . 6 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
15	14	anbi1i 458	. . . . 5 |- ( ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) <-> ( ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) /\ x R y ) )
16	10, 15	sylibr 134	. . . 4 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
17	16	eximi 1646	. . 3 |- ( E. x ( ( x e. A /\ x R y ) /\ y e. B ) -> E. x ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
18	1	elima2 5074	. . . . 5 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
19	18	anbi1i 458	. . . 4 |- ( ( y e. ( R " A ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
20		elin 3387	. . . 4 |- ( y e. ( ( R " A ) i^i B ) <-> ( y e. ( R " A ) /\ y e. B ) )
21		19.41v 1949	. . . 4 |- ( E. x ( ( x e. A /\ x R y ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
22	19, 20, 21	3bitr4i 212	. . 3 |- ( y e. ( ( R " A ) i^i B ) <-> E. x ( ( x e. A /\ x R y ) /\ y e. B ) )
23	1	elima2 5074	. . 3 |- ( y e. ( R " ( A i^i ( `' R " B ) ) ) <-> E. x ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
24	17, 22, 23	3imtr4i 201	. 2 |- ( y e. ( ( R " A ) i^i B ) -> y e. ( R " ( A i^i ( `' R " B ) ) ) )
25	24	ssriv 3228	1 |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Syntax hints:   /\ wa 104   E.wex 1538   e. wcel 2200   i^i cin 3196   C_ wss 3197   class class class wbr 4083   `'ccnv 4718   "cima 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by: (None)