Theorem dminss 5547

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 2052	. . . . . . 7 |- ( ( x e. A /\ x R y ) -> E. x ( x e. A /\ x R y ) )
2	1	ancoms 469	. . . . . 6 |- ( ( x R y /\ x e. A ) -> E. x ( x e. A /\ x R y ) )
3		vex 3203	. . . . . . 7 |- y e. _V
4	3	elima2 5472	. . . . . 6 |- ( y e. ( R " A ) <-> E. x ( x e. A /\ x R y ) )
5	2, 4	sylibr 224	. . . . 5 |- ( ( x R y /\ x e. A ) -> y e. ( R " A ) )
6		simpl 473	. . . . . 6 |- ( ( x R y /\ x e. A ) -> x R y )
7		vex 3203	. . . . . . 7 |- x e. _V
8	3, 7	brcnv 5305	. . . . . 6 |- ( y `' R x <-> x R y )
9	6, 8	sylibr 224	. . . . 5 |- ( ( x R y /\ x e. A ) -> y `' R x )
10	5, 9	jca 554	. . . 4 |- ( ( x R y /\ x e. A ) -> ( y e. ( R " A ) /\ y `' R x ) )
11	10	eximi 1762	. . 3 |- ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12	7	eldm 5321	. . . . 5 |- ( x e. dom R <-> E. y x R y )
13	12	anbi1i 731	. . . 4 |- ( ( x e. dom R /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
14		elin 3796	. . . 4 |- ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15		19.41v 1914	. . . 4 |- ( E. y ( x R y /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
16	13, 14, 15	3bitr4i 292	. . 3 |- ( x e. ( dom R i^i A ) <-> E. y ( x R y /\ x e. A ) )
17	7	elima2 5472	. . 3 |- ( x e. ( `' R " ( R " A ) ) <-> E. y ( y e. ( R " A ) /\ y `' R x ) )
18	11, 16, 17	3imtr4i 281	. 2 |- ( x e. ( dom R i^i A ) -> x e. ( `' R " ( R " A ) ) )
19	18	ssriv 3607	1 |- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Syntax hints:   /\ wa 384   E.wex 1704   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  lmhmlsp  19049  cnclsi  21076  kgencn3  21361  kqsat  21534  kqcldsat  21536  cfilucfil  22364