Theorem dmcosseq 4603

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 4601	. . 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 2939	. . . . . . . 8 |- ( ran B C_ dom A -> ( y e. ran B -> y e. dom A ) )
4		vex 2560	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 4577	. . . . . . . . . 10 |- ( y e. ran B <-> E. x x B y )
6	4	eldm 4532	. . . . . . . . . 10 |- ( y e. dom A <-> E. z y A z )
7	5, 6	imbi12i 228	. . . . . . . . 9 |- ( ( y e. ran B -> y e. dom A ) <-> ( E. x x B y -> E. z y A z ) )
8		19.8a 1482	. . . . . . . . . . 11 |- ( x B y -> E. x x B y )
9	8	imim1i 54	. . . . . . . . . 10 |- ( ( E. x x B y -> E. z y A z ) -> ( x B y -> E. z y A z ) )
10		pm3.2 126	. . . . . . . . . . 11 |- ( x B y -> ( y A z -> ( x B y /\ y A z ) ) )
11	10	eximdv 1760	. . . . . . . . . 10 |- ( x B y -> ( E. z y A z -> E. z ( x B y /\ y A z ) ) )
12	9, 11	sylcom 25	. . . . . . . . 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 114	. . . . . . . 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 1760	. . . . . 6 |- ( ran B C_ dom A -> ( E. y x B y -> E. y E. z ( x B y /\ y A z ) ) )
16		excom 1554	. . . . . 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 151	. . . . 5 |- ( ran B C_ dom A -> ( E. y x B y -> E. z E. y ( x B y /\ y A z ) ) )
18		vex 2560	. . . . . . 7 |- x e. _V
19		vex 2560	. . . . . . 7 |- z e. _V
20	18, 19	opelco 4507	. . . . . 6 |- ( <. x , z >. e. ( A o. B ) <-> E. y ( x B y /\ y A z ) )
21	20	exbii 1496	. . . . 5 |- ( E. z <. x , z >. e. ( A o. B ) <-> E. z E. y ( x B y /\ y A z ) )
22	17, 21	syl6ibr 151	. . . 4 |- ( ran B C_ dom A -> ( E. y x B y -> E. z <. x , z >. e. ( A o. B ) ) )
23	18	eldm 4532	. . . 4 |- ( x e. dom B <-> E. y x B y )
24	18	eldm2 4533	. . . 4 |- ( x e. dom ( A o. B ) <-> E. z <. x , z >. e. ( A o. B ) )
25	22, 23, 24	3imtr4g 194	. . 3 |- ( ran B C_ dom A -> ( x e. dom B -> x e. dom ( A o. B ) ) )
26	25	ssrdv 2951	. 2 |- ( ran B C_ dom A -> dom B C_ dom ( A o. B ) )
27	2, 26	eqssd 2962	1 |- ( ran B C_ dom A -> dom ( A o. B ) = dom B )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   <.cop 3378   class class class wbr 3764   dom cdm 4345   ran crn 4346   o. ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356

This theorem is referenced by:  dmcoeq  4604  fnco  5007