Theorem dmcosseq 5274

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 5272	. . 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 3493	. . . . . . . 8 |- ( ran B C_ dom A -> ( y e. ran B -> y e. dom A ) )
4		vex 3112	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 5253	. . . . . . . . . 10 |- ( y e. ran B <-> E. x x B y )
6	4	eldm 5210	. . . . . . . . . 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 1858	. . . . . . . . . . 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 1711	. . . . . . . . . 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 1711	. . . . . 6 |- ( ran B C_ dom A -> ( E. y x B y -> E. y E. z ( x B y /\ y A z ) ) )
16		excom 1850	. . . . . 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 3112	. . . . . . 7 |- x e. _V
19		vex 3112	. . . . . . 7 |- z e. _V
20	18, 19	opelco 5184	. . . . . 6 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
21	20	exbii 1668	. . . . 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 5210	. . . 4 |- ( x e. dom B <-> E. y x B y )
24	18	eldm2 5211	. . . 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 3505	. 2 |- ( ran B C_ dom A -> dom B C_ dom ( A o. B ) )
27	2, 26	eqssd 3516	1 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   C_ wss 3471   <.cop 4038   class class class wbr 4456   dom cdm 5008   ran crn 5009   o. ccom 5012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019

This theorem is referenced by:  dmcoeq  5275  fnco  5695  dvsinax  31869  fnresfnco  32372