Theorem dmcosseqOLD 4215

Description: Domain of a composition.

Assertion

Ref	Expression
dmcosseqOLD	|- (ran B C_ dom A -> dom ( A o. B) = dom B)

Proof of Theorem dmcosseqOLD

Step	Hyp	Ref	Expression
1		hbe1 1363	. . . . . . . 8 |- (E.x xBz -> A.xE.x xBz)
2		ax-17 1317	. . . . . . . 8 |- (E.y zAy -> A.xE.y zAy)
3	1, 2	hbim 1354	. . . . . . 7 |- ((E.x xBz -> E.y zAy) -> A.x(E.x xBz -> E.y zAy))
4	3	hbal 1352	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> A.xA.z(E.x xBz -> E.y zAy))
5		hba1 1350	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> A.zA.z(E.x xBz -> E.y zAy))
6		19.8a 1376	. . . . . . . . . 10 |- (xBz -> E.x xBz)
7	6	imim1i 19	. . . . . . . . 9 |- ((E.x xBz -> E.y zAy) -> (xBz -> E.y zAy))
8	7	ancld 322	. . . . . . . 8 |- ((E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
9	8	a4s 1330	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
10	5, 9	eximd 1410	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> (E.z xBz -> E.z(xBz /\ E.y zAy)))
11	4, 10	19.21ai 1345	. . . . 5 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z xBz -> E.z(xBz /\ E.y zAy)))
12		exsimpl 1461	. . . . . 6 |- (E.z(xBz /\ E.y zAy) -> E.z xBz)
13	12	ax-gen 1305	. . . . 5 |- A.x(E.z(xBz /\ E.y zAy) -> E.z xBz)
14	11, 13	jctil 316	. . . 4 |- (A.z(E.x xBz -> E.y zAy) -> (A.x(E.z(xBz /\ E.y zAy) -> E.z xBz) /\ A.x(E.z xBz -> E.z(xBz /\ E.y zAy))))
15		albiim 1465	. . . 4 |- (A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz) <-> (A.x(E.z(xBz /\ E.y zAy) -> E.z xBz) /\ A.x(E.z xBz -> E.z(xBz /\ E.y zAy))))
16	14, 15	sylibr 217	. . 3 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz))
17		visset 2295	. . . . . 6 |- z e. _V
18	17	elrn 4197	. . . . 5 |- (z e. ran B <-> E.x xBz)
19	17	eldm 4153	. . . . 5 |- (z e. dom A <-> E.y zAy)
20	18, 19	imbi12i 205	. . . 4 |- ((z e. ran B -> z e. dom A) <-> (E.x xBz -> E.y zAy))
21	20	albii 1346	. . 3 |- (A.z(z e. ran B -> z e. dom A) <-> A.z(E.x xBz -> E.y zAy))
22		visset 2295	. . . . . . 7 |- x e. _V
23	22	eldm2 4154	. . . . . 6 |- (x e. dom ( A o. B) <-> E.y<.x, y>. e. (A o. B))
24		visset 2295	. . . . . . . 8 |- y e. _V
25	22, 24	opelco 4130	. . . . . . 7 |- (<.x, y>. e. (A o. B) <-> E.z(xBz /\ zAy))
26	25	exbii 1398	. . . . . 6 |- (E.y<.x, y>. e. (A o. B) <-> E.yE.z(xBz /\ zAy))
27		excom 1393	. . . . . . 7 |- (E.yE.z(xBz /\ zAy) <-> E.zE.y(xBz /\ zAy))
28		19.42v 1688	. . . . . . . 8 |- (E.y(xBz /\ zAy) <-> (xBz /\ E.y zAy))
29	28	exbii 1398	. . . . . . 7 |- (E.zE.y(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
30	27, 29	bitri 190	. . . . . 6 |- (E.yE.z(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
31	23, 26, 30	3bitri 194	. . . . 5 |- (x e. dom ( A o. B) <-> E.z(xBz /\ E.y zAy))
32	22	eldm 4153	. . . . 5 |- (x e. dom B <-> E.z xBz)
33	31, 32	bibi12i 672	. . . 4 |- ((x e. dom ( A o. B) <-> x e. dom B) <-> (E.z(xBz /\ E.y zAy) <-> E.z xBz))
34	33	albii 1346	. . 3 |- (A.x(x e. dom ( A o. B) <-> x e. dom B) <-> A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz))
35	16, 21, 34	3imtr4i 236	. 2 |- (A.z(z e. ran B -> z e. dom A) -> A.x(x e. dom ( A o. B) <-> x e. dom B))
36		dfss2 2610	. 2 |- (ran B C_ dom A <-> A.z(z e. ran B -> z e. dom A))
37		dfcleq 1878	. 2 |- (dom ( A o. B) = dom B <-> A.x(x e. dom ( A o. B) <-> x e. dom B))
38	35, 36, 37	3imtr4i 236	1 |- (ran B C_ dom A -> dom ( A o. B) = dom B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = 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 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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