Theorem dminss 5182

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 1639	. . . . . . 7 |- ( ( x e. A /\ x R y ) -> E. x ( x e. A /\ x R y ) )
2	1	ancoms 268	. . . . . 6 |- ( ( x R y /\ x e. A ) -> E. x ( x e. A /\ x R y ) )
3		vex 2818	. . . . . . 7 |- y e. _V
4	3	elima2 5112	. . . . . 6 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
5	2, 4	sylibr 134	. . . . 5 |- ( ( x R y /\ x e. A ) -> y e. ( R " A ) )
6		simpl 109	. . . . . 6 |- ( ( x R y /\ x e. A ) -> x R y )
7		vex 2818	. . . . . . 7 |- x e. _V
8	3, 7	brcnv 4943	. . . . . 6 |- ( y `' R x <-> x R y )
9	6, 8	sylibr 134	. . . . 5 |- ( ( x R y /\ x e. A ) -> y `' R x )
10	5, 9	jca 306	. . . 4 |- ( ( x R y /\ x e. A ) -> ( y e. ( R " A ) /\ y `' R x ) )
11	10	eximi 1649	. . 3 |- ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12	7	eldm 4958	. . . . 5 |- ( x e. dom R <-> E. y x R y )
13	12	anbi1i 458	. . . 4 |- ( ( x e. dom R /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
14		elin 3406	. . . 4 |- ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15		19.41v 1954	. . . 4 |- ( E. y ( x R y /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
16	13, 14, 15	3bitr4i 212	. . 3 |- ( x e. ( dom R i^i A ) <-> E. y ( x R y /\ x e. A ) )
17	7	elima2 5112	. . 3 |- ( x e. ( `' R " ( R " A ) ) <-> E. y ( y e. ( R " A ) /\ y `' R x ) )
18	11, 16, 17	3imtr4i 201	. 2 |- ( x e. ( dom R i^i A ) -> x e. ( `' R " ( R " A ) ) )
19	18	ssriv 3246	1 |- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Syntax hints:   /\ wa 104   E.wex 1541   e. wcel 2205   i^i cin 3213   C_ wss 3214   class class class wbr 4114   `'ccnv 4753   dom cdm 4754   "cima 4757

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by: (None)