Theorem dminss 4948

Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising." (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
dminss	|- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Proof of Theorem dminss

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

Step	Hyp	Ref	Expression
1		19.8a 1569	. . . . . . 7 |- ( ( x e. A /\ x R y ) -> E. x ( x e. A /\ x R y ) )
2	1	ancoms 266	. . . . . 6 |- ( ( x R y /\ x e. A ) -> E. x ( x e. A /\ x R y ) )
3		vex 2684	. . . . . . 7 |- y e. _V
4	3	elima2 4882	. . . . . 6 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
5	2, 4	sylibr 133	. . . . 5 |- ( ( x R y /\ x e. A ) -> y e. ( R " A ) )
6		simpl 108	. . . . . 6 |- ( ( x R y /\ x e. A ) -> x R y )
7		vex 2684	. . . . . . 7 |- x e. _V
8	3, 7	brcnv 4717	. . . . . 6 |- ( y `' R x <-> x R y )
9	6, 8	sylibr 133	. . . . 5 |- ( ( x R y /\ x e. A ) -> y `' R x )
10	5, 9	jca 304	. . . 4 |- ( ( x R y /\ x e. A ) -> ( y e. ( R " A ) /\ y `' R x ) )
11	10	eximi 1579	. . 3 |- ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12	7	eldm 4731	. . . . 5 |- ( x e. dom R <-> E. y x R y )
13	12	anbi1i 453	. . . 4 |- ( ( x e. dom R /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
14		elin 3254	. . . 4 |- ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15		19.41v 1874	. . . 4 |- ( E. y ( x R y /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
16	13, 14, 15	3bitr4i 211	. . 3 |- ( x e. ( dom R i^i A ) <-> E. y ( x R y /\ x e. A ) )
17	7	elima2 4882	. . 3 |- ( x e. ( `' R " ( R " A ) ) <-> E. y ( y e. ( R " A ) /\ y `' R x ) )
18	11, 16, 17	3imtr4i 200	. 2 |- ( x e. ( dom R i^i A ) -> x e. ( `' R " ( R " A ) ) )
19	18	ssriv 3096	1 |- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Syntax hints:   /\ wa 103   E.wex 1468   e. wcel 1480   i^i cin 3065   C_ wss 3066   class class class wbr 3924   `'ccnv 4533   dom cdm 4534   "cima 4537

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by: (None)