Theorem dmcosseq 4214

Description: Domain of a composition. (The proof was 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

Step	Hyp	Ref	Expression
1		dmcoss 4211	. . 3 |- dom ( A o. B) C_ dom B
2	1	a1i 8	. 2 |- (ran B C_ dom A -> dom ( A o. B) C_ dom B)
3		ssel 2615	. . . . . . . 8 |- (ran B C_ dom A -> (y e. ran B -> y e. dom A))
4		visset 2295	. . . . . . . . . . 11 |- y e. _V
5	4	elrn 4197	. . . . . . . . . 10 |- (y e. ran B <-> E.x xBy)
6	4	eldm 4153	. . . . . . . . . 10 |- (y e. dom A <-> E.z yAz)
7	5, 6	imbi12i 205	. . . . . . . . 9 |- ((y e. ran B -> y e. dom A) <-> (E.x xBy -> E.z yAz))
8		19.8a 1376	. . . . . . . . . . 11 |- (xBy -> E.x xBy)
9	8	imim1i 19	. . . . . . . . . 10 |- ((E.x xBy -> E.z yAz) -> (xBy -> E.z yAz))
10		pm3.2 305	. . . . . . . . . . 11 |- (xBy -> (yAz -> (xBy /\ yAz)))
11	10	eximdv 1669	. . . . . . . . . 10 |- (xBy -> (E.z yAz -> E.z(xBy /\ yAz)))
12	9, 11	sylcom 62	. . . . . . . . 9 |- ((E.x xBy -> E.z yAz) -> (xBy -> E.z(xBy /\ yAz)))
13	7, 12	sylbi 216	. . . . . . . 8 |- ((y e. ran B -> y e. dom A) -> (xBy -> E.z(xBy /\ yAz)))
14	3, 13	syl 12	. . . . . . 7 |- (ran B C_ dom A -> (xBy -> E.z(xBy /\ yAz)))
15	14	eximdv 1669	. . . . . 6 |- (ran B C_ dom A -> (E.y xBy -> E.yE.z(xBy /\ yAz)))
16		excom 1393	. . . . . 6 |- (E.zE.y(xBy /\ yAz) <-> E.yE.z(xBy /\ yAz))
17	15, 16	syl6ibr 230	. . . . 5 |- (ran B C_ dom A -> (E.y xBy -> E.zE.y(xBy /\ yAz)))
18		visset 2295	. . . . . . 7 |- x e. _V
19		visset 2295	. . . . . . 7 |- z e. _V
20	18, 19	opelco 4130	. . . . . 6 |- (<.x, z>. e. (A o. B) <-> E.y(xBy /\ yAz))
21	20	exbii 1398	. . . . 5 |- (E.z<.x, z>. e. (A o. B) <-> E.zE.y(xBy /\ yAz))
22	17, 21	syl6ibr 230	. . . 4 |- (ran B C_ dom A -> (E.y xBy -> E.z<.x, z>. e. (A o. B)))
23	18	eldm 4153	. . . 4 |- (x e. dom B <-> E.y xBy)
24	18	eldm2 4154	. . . 4 |- (x e. dom ( A o. B) <-> E.z<.x, z>. e. (A o. B))
25	22, 23, 24	3imtr4g 612	. . 3 |- (ran B C_ dom A -> (x e. dom B -> x e. dom ( A o. B)))
26	25	ssrdv 2622	. 2 |- (ran B C_ dom A -> dom B C_ dom ( A o. B))
27	2, 26	eqssd 2633	1 |- (ran B C_ dom A -> dom ( A o. B) = dom B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   C_ wss 2593  <.cop 3046   class class class wbr 3338  dom cdm 3986  ran crn 3987   o. ccom 3990

This theorem is referenced by:  dmcoeq 4216  fnco 4521  fcoOLD 4574  cncfmet1 9184

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005

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