Theorem dmcoss 5047

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 1549	. . . 4 |- F/ y E. y x B y
2		exsimpl 1670	. . . . 5 |- ( E. z ( x B z /\ z A y ) -> E. z x B z )
3		vex 2824	. . . . . 6 |- x e. _V
4		vex 2824	. . . . . 6 |- y e. _V
5	3, 4	opelco 4947	. . . . 5 |- ( <. x , y >. e. ( A o. B ) <-> E. z ( x B z /\ z A y ) )
6		breq2 4129	. . . . . 6 |- ( y = z -> ( x B y <-> x B z ) )
7	6	cbvexv 1974	. . . . 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 1647	. . 3 |- ( E. y <. x , y >. e. ( A o. B ) -> E. y x B y )
10	3	eldm2 4974	. . 3 |- ( x e. dom ( A o. B ) <-> E. y <. x , y >. e. ( A o. B ) )
11	3	eldm 4973	. . 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 3252	1 |- dom ( A o. B ) C_ dom B

Syntax hints:   /\ wa 104   E.wex 1545   e. wcel 2209   C_ wss 3220   <.cop 3708   class class class wbr 4125   dom cdm 4769   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-co 4778  df-dm 4779

This theorem is referenced by:  rncoss  5048  dmcosseq  5049  cossxp  5305  funco  5412  cofunexg  6328  casefun  7415  djufun  7434  ctssdccl  7441  znleval  14960