Theorem dminss 5097

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 1613	. . . . . . 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 2775	. . . . . . 7 |- y e. _V
4	3	elima2 5028	. . . . . 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 2775	. . . . . . 7 |- x e. _V
8	3, 7	brcnv 4861	. . . . . 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 1623	. . 3 |- ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12	7	eldm 4875	. . . . 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 3356	. . . 4 |- ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15		19.41v 1926	. . . 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 5028	. . 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 3197	1 |- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Syntax hints:   /\ wa 104   E.wex 1515   e. wcel 2176   i^i cin 3165   C_ wss 3166   class class class wbr 4044   `'ccnv 4674   dom cdm 4675   "cima 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by: (None)