Theorem dmcosseq 4857

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 4855	. . 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 3122	. . . . . . . 8 |- ( ran B C_ dom A -> ( y e. ran B -> y e. dom A ) )
4		vex 2715	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 4829	. . . . . . . . . 10 |- ( y e. ran B <-> E. x x B y )
6	4	eldm 4783	. . . . . . . . . 10 |- ( y e. dom A <-> E. z y A z )
7	5, 6	imbi12i 238	. . . . . . . . 9 |- ( ( y e. ran B -> y e. dom A ) <-> ( E. x x B y -> E. z y A z ) )
8		19.8a 1570	. . . . . . . . . . 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		pm3.2 138	. . . . . . . . . . 11 |- ( x B y -> ( y A z -> ( x B y /\ y A z ) ) )
11	10	eximdv 1860	. . . . . . . . . 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 120	. . . . . . . 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 1860	. . . . . 6 |- ( ran B C_ dom A -> ( E. y x B y -> E. y E. z ( x B y /\ y A z ) ) )
16		excom 1644	. . . . . 6 |- ( E. z E. y ( x B y /\ y A z ) <-> E. y E. z ( x B y /\ y A z ) )
17	15, 16	syl6ibr 161	. . . . 5 |- ( ran B C_ dom A -> ( E. y x B y -> E. z E. y ( x B y /\ y A z ) ) )
18		vex 2715	. . . . . . 7 |- x e. _V
19		vex 2715	. . . . . . 7 |- z e. _V
20	18, 19	opelco 4758	. . . . . 6 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
21	20	exbii 1585	. . . . 5 |- ( E. z <. x , z >. e. ( A o. B ) <-> E. z E. y ( x B y /\ y A z ) )
22	17, 21	syl6ibr 161	. . . 4 |- ( ran B C_ dom A -> ( E. y x B y -> E. z <. x , z >. e. ( A o. B ) ) )
23	18	eldm 4783	. . . 4 |- ( x e. dom B <-> E. y x B y )
24	18	eldm2 4784	. . . 4 |- ( x e. dom ( A o. B ) <-> E. z <. x , z >. e. ( A o. B ) )
25	22, 23, 24	3imtr4g 204	. . 3 |- ( ran B C_ dom A -> ( x e. dom B -> x e. dom ( A o. B ) ) )
26	25	ssrdv 3134	. 2 |- ( ran B C_ dom A -> dom B C_ dom ( A o. B ) )
27	2, 26	eqssd 3145	1 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   E.wex 1472   e. wcel 2128   C_ wss 3102   <.cop 3563   class class class wbr 3965   dom cdm 4586   ran crn 4587   o. ccom 4590

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597

This theorem is referenced by:  dmcoeq  4858  fnco  5278