Theorem imainss 4847

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 2622	. . . . . . . . . . 11 |- y e. _V
2		vex 2622	. . . . . . . . . . 11 |- x e. _V
3	1, 2	brcnv 4619	. . . . . . . . . 10 |- ( y `' R x <-> x R y )
4		19.8a 1527	. . . . . . . . . 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 498	. . . . . . 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 3183	. . . . . . 7 |- ( x e. ( A i^i ( `' R " B ) ) <-> ( x e. A /\ x e. ( `' R " B ) ) )
12	2	elima2 4780	. . . . . . . 8 |- ( x e. ( `' R " B ) <-> E. y ( y e. B /\ y `' R x ) )
13	12	anbi2i 445	. . . . . . 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 446	. . . . 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 1536	. . 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 4780	. . . . 5 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
19	18	anbi1i 446	. . . 4 |- ( ( y e. ( R " A ) /\ y e. B ) <-> ( E. x ( x e. A /\ x R y ) /\ y e. B ) )
20		elin 3183	. . . 4 |- ( y e. ( ( R " A ) i^i B ) <-> ( y e. ( R " A ) /\ y e. B ) )
21		19.41v 1830	. . . 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 4780	. . 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 3029	1 |- ( ( R " A ) i^i B ) C_ ( R " ( A i^i ( `' R " B ) ) )

Syntax hints:   /\ wa 102   E.wex 1426   e. wcel 1438   i^i cin 2998   C_ wss 2999   class class class wbr 3845   `'ccnv 4437   "cima 4441

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451

This theorem is referenced by: (None)