Theorem dmcoss 5000

Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmcoss	|- dom ( A o. B ) C_ dom B

Proof of Theorem dmcoss

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

Step	Hyp	Ref	Expression
1		nfe1 1542	. . . 4 |- F/ y E. y x B y
2		exsimpl 1663	. . . . 5 |- ( E. z ( x B z /\ z A y ) -> E. z x B z )
3		vex 2803	. . . . . 6 |- x e. _V
4		vex 2803	. . . . . 6 |- y e. _V
5	3, 4	opelco 4900	. . . . 5 |- ( <. x , y >. e. ( A o. B ) <-> E. z ( x B z /\ z A y ) )
6		breq2 4090	. . . . . 6 |- ( y = z -> ( x B y <-> x B z ) )
7	6	cbvexv 1965	. . . . 5 |- ( E. y x B y <-> E. z x B z )
8	2, 5, 7	3imtr4i 201	. . . 4 |- ( <. x , y >. e. ( A o. B ) -> E. y x B y )
9	1, 8	exlimi 1640	. . 3 |- ( E. y <. x , y >. e. ( A o. B ) -> E. y x B y )
10	3	eldm2 4927	. . 3 |- ( x e. dom ( A o. B ) <-> E. y <. x , y >. e. ( A o. B ) )
11	3	eldm 4926	. . 3 |- ( x e. dom B <-> E. y x B y )
12	9, 10, 11	3imtr4i 201	. 2 |- ( x e. dom ( A o. B ) -> x e. dom B )
13	12	ssriv 3229	1 |- dom ( A o. B ) C_ dom B

Syntax hints:   /\ wa 104   E.wex 1538   e. wcel 2200   C_ wss 3198   <.cop 3670   class class class wbr 4086   dom cdm 4723   o. ccom 4727

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-co 4732  df-dm 4733

This theorem is referenced by:  rncoss  5001  dmcosseq  5002  cossxp  5257  funco  5364  cofunexg  6266  casefun  7275  djufun  7294  ctssdccl  7301  znleval  14657