Theorem dminss 5025

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 1583	. . . . . . 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 2733	. . . . . . 7 |- y e. _V
4	3	elima2 4959	. . . . . 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 2733	. . . . . . 7 |- x e. _V
8	3, 7	brcnv 4794	. . . . . 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 1593	. . 3 |- ( E. y ( x R y /\ x e. A ) -> E. y ( y e. ( R " A ) /\ y `' R x ) )
12	7	eldm 4808	. . . . 5 |- ( x e. dom R <-> E. y x R y )
13	12	anbi1i 455	. . . 4 |- ( ( x e. dom R /\ x e. A ) <-> ( E. y x R y /\ x e. A ) )
14		elin 3310	. . . 4 |- ( x e. ( dom R i^i A ) <-> ( x e. dom R /\ x e. A ) )
15		19.41v 1895	. . . 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 4959	. . 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 3151	1 |- ( dom R i^i A ) C_ ( `' R " ( R " A ) )

Syntax hints:   /\ wa 103   E.wex 1485   e. wcel 2141   i^i cin 3120   C_ wss 3121   class class class wbr 3989   `'ccnv 4610   dom cdm 4611   "cima 4614

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by: (None)