Theorem dmcosseq 4933

Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmcosseq	|- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Proof of Theorem dmcosseq

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

Step	Hyp	Ref	Expression
1		dmcoss 4931	. . 3 |- dom ( A o. B ) C_ dom B
2	1	a1i 9	. 2 |- ( ran B C_ dom A -> dom ( A o. B ) C_ dom B )
3		ssel 3173	. . . . . . . 8 |- ( ran B C_ dom A -> ( y e. ran B -> y e. dom A ) )
4		vex 2763	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 4905	. . . . . . . . . 10 |- ( y e. ran B <-> E. x x B y )
6	4	eldm 4859	. . . . . . . . . 10 |- ( y e. dom A <-> E. z y A z )
7	5, 6	imbi12i 239	. . . . . . . . 9 |- ( ( y e. ran B -> y e. dom A ) <-> ( E. x x B y -> E. z y A z ) )
8		19.8a 1601	. . . . . . . . . . 11 |- ( x B y -> E. x x B y )
9	8	imim1i 60	. . . . . . . . . 10 |- ( ( E. x x B y -> E. z y A z ) -> ( x B y -> E. z y A z ) )
10		pm3.2 139	. . . . . . . . . . 11 |- ( x B y -> ( y A z -> ( x B y /\ y A z ) ) )
11	10	eximdv 1891	. . . . . . . . . 10 |- ( x B y -> ( E. z y A z -> E. z ( x B y /\ y A z ) ) )
12	9, 11	sylcom 28	. . . . . . . . 9 |- ( ( E. x x B y -> E. z y A z ) -> ( x B y -> E. z ( x B y /\ y A z ) ) )
13	7, 12	sylbi 121	. . . . . . . 8 |- ( ( y e. ran B -> y e. dom A ) -> ( x B y -> E. z ( x B y /\ y A z ) ) )
14	3, 13	syl 14	. . . . . . 7 |- ( ran B C_ dom A -> ( x B y -> E. z ( x B y /\ y A z ) ) )
15	14	eximdv 1891	. . . . . 6 |- ( ran B C_ dom A -> ( E. y x B y -> E. y E. z ( x B y /\ y A z ) ) )
16		excom 1675	. . . . . 6 |- ( E. z E. y ( x B y /\ y A z ) <-> E. y E. z ( x B y /\ y A z ) )
17	15, 16	imbitrrdi 162	. . . . 5 |- ( ran B C_ dom A -> ( E. y x B y -> E. z E. y ( x B y /\ y A z ) ) )
18		vex 2763	. . . . . . 7 |- x e. _V
19		vex 2763	. . . . . . 7 |- z e. _V
20	18, 19	opelco 4834	. . . . . 6 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
21	20	exbii 1616	. . . . 5 |- ( E. z <. x , z >. e. ( A o. B ) <-> E. z E. y ( x B y /\ y A z ) )
22	17, 21	imbitrrdi 162	. . . 4 |- ( ran B C_ dom A -> ( E. y x B y -> E. z <. x , z >. e. ( A o. B ) ) )
23	18	eldm 4859	. . . 4 |- ( x e. dom B <-> E. y x B y )
24	18	eldm2 4860	. . . 4 |- ( x e. dom ( A o. B ) <-> E. z <. x , z >. e. ( A o. B ) )
25	22, 23, 24	3imtr4g 205	. . 3 |- ( ran B C_ dom A -> ( x e. dom B -> x e. dom ( A o. B ) ) )
26	25	ssrdv 3185	. 2 |- ( ran B C_ dom A -> dom B C_ dom ( A o. B ) )
27	2, 26	eqssd 3196	1 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   C_ wss 3153   <.cop 3621   class class class wbr 4029   dom cdm 4659   ran crn 4660   o. ccom 4663

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670

This theorem is referenced by:  dmcoeq  4934  fnco  5362