Theorem imainss 4962

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 2692	. . . . . . . . . . 11 |- y e. _V
2		vex 2692	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4730	. . . . . . . . . 10 |- ( y `' R x <-> x R y )
4		19.8a 1570	. . . . . . . . . 10 |- ( ( y e. B /\ y `' R x ) -> E. y ( y e. B /\ y `' R x ) )
5	3, 4	sylan2br 286	. . . . . . . . 9 |- ( ( y e. B /\ x R y ) -> E. y ( y e. B /\ y `' R x ) )
6	5	ancoms 266	. . . . . . . 8 |- ( ( x R y /\ y e. B ) -> E. y ( y e. B /\ y `' R x ) )
7	6	anim2i 340	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
8		simprl 521	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> x R y )
9	7, 8	jca 304	. . . . . 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 398	. . . . 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 3264	. . . . . . 7 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ x e. ( `' R " B ) ) )
12	2	elima2 4895	. . . . . . . 8 |- ( x e. ( `' R " B ) <-> E. y ( y e. B /\ y `' R x ) )
13	12	anbi2i 453	. . . . . . 7 |- ( ( x e. A /\ x e. ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
14	11, 13	bitri 183	. . . . . 6 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
15	14	anbi1i 454	. . . . 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 133	. . . 4 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
17	16	eximi 1580	. . 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 4895	. . . . 5 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
19	18	anbi1i 454	. . . 4 |- ( ( y e. ( R " A ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
20		elin 3264	. . . 4 |- ( y e. ( ( R " A ) i^i B ) <-> ( y e. ( R " A ) /\ y e. B ) )
21		19.41v 1875	. . . 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 211	. . 3 |- ( y e. ( ( R " A ) i^i B ) <-> E. x ( ( x e. A /\ x R y ) /\ y e. B ) )
23	1	elima2 4895	. . 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 200	. 2 |- ( y e. ( ( R " A ) i^i B ) -> y e. ( R " ( A i^i ( `' R " B ) ) ) )
25	24	ssriv 3106	1 |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Syntax hints:   /\ wa 103   E.wex 1469   e. wcel 1481   i^i cin 3075   C_ wss 3076   class class class wbr 3937   `'ccnv 4546   "cima 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)