Theorem dmcosseq 5199

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 5197	. . 3 |- dom ( A o. B ) C_ dom B
2	1	a1i 11	. 2 |- ( ran B C_ dom A -> dom ( A o. B ) C_ dom B )
3		ssel 3448	. . . . . . . 8 |- ( ran B C_ dom A -> ( y e. ran B -> y e. dom A ) )
4		vex 3071	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 5178	. . . . . . . . . 10 |- ( y e. ran B <-> E. x x B y )
6	4	eldm 5135	. . . . . . . . . 10 |- ( y e. dom A <-> E. z y A z )
7	5, 6	imbi12i 326	. . . . . . . . 9 |- ( ( y e. ran B -> y e. dom A ) <-> ( E. x x B y -> E. z y A z ) )
8		19.8a 1795	. . . . . . . . . . 11 |- ( x B y -> E. x x B y )
9	8	imim1i 58	. . . . . . . . . 10 |- ( ( E. x x B y -> E. z y A z ) -> ( x B y -> E. z y A z ) )
10		pm3.2 447	. . . . . . . . . . 11 |- ( x B y -> ( y A z -> ( x B y /\ y A z ) ) )
11	10	eximdv 1677	. . . . . . . . . 10 |- ( x B y -> ( E. z y A z -> E. z ( x B y /\ y A z ) ) )
12	9, 11	sylcom 29	. . . . . . . . 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 195	. . . . . . . 8 |- ( ( y e. ran B -> y e. dom A ) -> ( x B y -> E. z ( x B y /\ y A z ) ) )
14	3, 13	syl 16	. . . . . . 7 |- ( ran B C_ dom A -> ( x B y -> E. z ( x B y /\ y A z ) ) )
15	14	eximdv 1677	. . . . . 6 |- ( ran B C_ dom A -> ( E. y x B y -> E. y E. z ( x B y /\ y A z ) ) )
16		excom 1789	. . . . . 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 227	. . . . 5 |- ( ran B C_ dom A -> ( E. y x B y -> E. z E. y ( x B y /\ y A z ) ) )
18		vex 3071	. . . . . . 7 |- x e. _V
19		vex 3071	. . . . . . 7 |- z e. _V
20	18, 19	opelco 5109	. . . . . 6 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
21	20	exbii 1635	. . . . 5 |- ( E. z <. x , z >. e. ( A o. B ) <-> E. z E. y ( x B y /\ y A z ) )
22	17, 21	syl6ibr 227	. . . 4 |- ( ran B C_ dom A -> ( E. y x B y -> E. z <. x , z >. e. ( A o. B ) ) )
23	18	eldm 5135	. . . 4 |- ( x e. dom B <-> E. y x B y )
24	18	eldm2 5136	. . . 4 |- ( x e. dom ( A o. B ) <-> E. z <. x , z >. e. ( A o. B ) )
25	22, 23, 24	3imtr4g 270	. . 3 |- ( ran B C_ dom A -> ( x e. dom B -> x e. dom ( A o. B ) ) )
26	25	ssrdv 3460	. 2 |- ( ran B C_ dom A -> dom B C_ dom ( A o. B ) )
27	2, 26	eqssd 3471	1 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   C_ wss 3426   <.cop 3981   class class class wbr 4390   dom cdm 4938   ran crn 4939   o. ccom 4942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949

This theorem is referenced by:  dmcoeq  5200  fnco  5617  fnresfnco  30170