Theorem dmcosseq 3361

Description: Domain of a composition.

Assertion

Ref	Expression
dmcosseq	|- (ran B (_ dom A -> dom ( A o. B) = dom B)

Proof of Theorem dmcosseq

Step	Hyp	Ref	Expression
1		hbe1 1015	. . . . . . . 8 |- (E.x xBz -> A.xE.x xBz)
2		ax-17 970	. . . . . . . 8 |- (E.y zAy -> A.xE.y zAy)
3	1, 2	hbim 1006	. . . . . . 7 |- ((E.x xBz -> E.y zAy) -> A.x(E.x xBz -> E.y zAy))
4	3	hbal 1004	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> A.xA.z(E.x xBz -> E.y zAy))
5		hba1 1002	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> A.zA.z(E.x xBz -> E.y zAy))
6		19.8a 1028	. . . . . . . . . 10 |- (xBz -> E.x xBz)
7	6	imim1i 16	. . . . . . . . 9 |- ((E.x xBz -> E.y zAy) -> (xBz -> E.y zAy))
8	7	ancld 298	. . . . . . . 8 |- ((E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
9	8	a4s 983	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
10	5, 9	19.22d 1061	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> (E.z xBz -> E.z(xBz /\ E.y zAy)))
11	4, 10	19.21ai 997	. . . . 5 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z xBz -> E.z(xBz /\ E.y zAy)))
12		pm3.26 319	. . . . . . 7 |- ((xBz /\ E.y zAy) -> xBz)
13	12	19.22i 1039	. . . . . 6 |- (E.z(xBz /\ E.y zAy) -> E.z xBz)
14	13	ax-gen 962	. . . . 5 |- A.x(E.z(xBz /\ E.y zAy) -> E.z xBz)
15	11, 14	jctil 292	. . . 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))))
16		albi 1106	. . . 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))))
17	15, 16	sylibr 200	. . 3 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz))
18		visset 1810	. . . . . 6 |- z e. V
19	18	elrn 3346	. . . . 5 |- (z e. ran B <-> E.x xBz)
20	18	eldm 3303	. . . . 5 |- (z e. dom A <-> E.y zAy)
21	19, 20	imbi12i 188	. . . 4 |- ((z e. ran B -> z e. dom A) <-> (E.x xBz -> E.y zAy))
22	21	albii 998	. . 3 |- (A.z(z e. ran B -> z e. dom A) <-> A.z(E.x xBz -> E.y zAy))
23		visset 1810	. . . . . . 7 |- x e. V
24	23	eldm2 3304	. . . . . 6 |- (x e. dom ( A o. B) <-> E.y<.x, y>. e. (A o. B))
25		visset 1810	. . . . . . . 8 |- y e. V
26	23, 25	opelco 3284	. . . . . . 7 |- (<.x, y>. e. (A o. B) <-> E.z(xBz /\ zAy))
27	26	exbii 1050	. . . . . 6 |- (E.y<.x, y>. e. (A o. B) <-> E.yE.z(xBz /\ zAy))
28		excom 1045	. . . . . . 7 |- (E.yE.z(xBz /\ zAy) <-> E.zE.y(xBz /\ zAy))
29		19.42v 1307	. . . . . . . 8 |- (E.y(xBz /\ zAy) <-> (xBz /\ E.y zAy))
30	29	exbii 1050	. . . . . . 7 |- (E.zE.y(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
31	28, 30	bitr 173	. . . . . 6 |- (E.yE.z(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
32	24, 27, 31	3bitr 177	. . . . 5 |- (x e. dom ( A o. B) <-> E.z(xBz /\ E.y zAy))
33	23	eldm 3303	. . . . 5 |- (x e. dom B <-> E.z xBz)
34	32, 33	bibi12i 609	. . . 4 |- ((x e. dom ( A o. B) <-> x e. dom B) <-> (E.z(xBz /\ E.y zAy) <-> E.z xBz))
35	34	albii 998	. . 3 |- (A.x(x e. dom ( A o. B) <-> x e. dom B) <-> A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz))
36	17, 22, 35	3imtr4 219	. 2 |- (A.z(z e. ran B -> z e. dom A) -> A.x(x e. dom ( A o. B) <-> x e. dom B))
37		dfss2 2055	. 2 |- (ran B (_ dom A <-> A.z(z e. ran B -> z e. dom A))
38		dfcleq 1469	. 2 |- (dom ( A o. B) = dom B <-> A.x(x e. dom ( A o. B) <-> x e. dom B))
39	36, 37, 38	3imtr4 219	1 |- (ran B (_ dom A -> dom ( A o. B) = dom B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979   (_ wss 2044  <.cop 2408   class class class wbr 2615  dom cdm 3166  ran crn 3167   o. ccom 3170

This theorem is referenced by:  dmcoeq 3362  fnco 3591  fco 3631  cncfmet1 7868

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185

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