Theorem imainss 4759

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 2604	. . . . . . . . . . 11 |- y e. _V
2		vex 2604	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4536	. . . . . . . . . 10 |- ( y `' R x <-> x R y )
4		19.8a 1522	. . . . . . . . . 10 |- ( ( y e. B /\ y `' R x ) -> E. y ( y e. B /\ y `' R x ) )
5	3, 4	sylan2br 282	. . . . . . . . 9 |- ( ( y e. B /\ x R y ) -> E. y ( y e. B /\ y `' R x ) )
6	5	ancoms 264	. . . . . . . 8 |- ( ( x R y /\ y e. B ) -> E. y ( y e. B /\ y `' R x ) )
7	6	anim2i 334	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
8		simprl 497	. . . . . . 7 |- ( ( x e. A /\ ( x R y /\ y e. B ) ) -> x R y )
9	7, 8	jca 300	. . . . . 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 392	. . . . 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 3155	. . . . . . 7 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ x e. ( `' R " B ) ) )
12	2	elima2 4694	. . . . . . . 8 |- ( x e. ( `' R " B ) <-> E. y ( y e. B /\ y `' R x ) )
13	12	anbi2i 444	. . . . . . 7 |- ( ( x e. A /\ x e. ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
14	11, 13	bitri 182	. . . . . 6 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ E. y ( y e. B /\ y `' R x ) ) )
15	14	anbi1i 445	. . . . 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 132	. . . 4 |- ( ( ( x e. A /\ x R y ) /\ y e. B ) -> ( x e. ( A i^i ( `' R " B ) ) /\ x R y ) )
17	16	eximi 1531	. . 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 4694	. . . . 5 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
19	18	anbi1i 445	. . . 4 |- ( ( y e. ( R " A ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
20		elin 3155	. . . 4 |- ( y e. ( ( R " A ) i^i B ) <-> ( y e. ( R " A ) /\ y e. B ) )
21		19.41v 1823	. . . 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 210	. . 3 |- ( y e. ( ( R " A ) i^i B ) <-> E. x ( ( x e. A /\ x R y ) /\ y e. B ) )
23	1	elima2 4694	. . 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 199	. 2 |- ( y e. ( ( R " A ) i^i B ) -> y e. ( R " ( A i^i ( `' R " B ) ) ) )
25	24	ssriv 3003	1 |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Syntax hints:   /\ wa 102   E.wex 1421   e. wcel 1433   i^i cin 2972   C_ wss 2973   class class class wbr 3785   `'ccnv 4362   "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: (None)