Theorem dmcosseq 3316

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 990	. . . . . . . 8 |- (E.x xBz -> A.xE.x xBz)
2		ax-17 1190	. . . . . . . 8 |- (E.y zAy -> A.xE.y zAy)
3	1, 2	hbim 983	. . . . . . 7 |- ((E.x xBz -> E.y zAy) -> A.x(E.x xBz -> E.y zAy))
4	3	hbal 981	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> A.xA.z(E.x xBz -> E.y zAy))
5		hba1 979	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> A.zA.z(E.x xBz -> E.y zAy))
6		19.8a 1005	. . . . . . . . . 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 960	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
10	5, 9	19.22d 1038	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> (E.z xBz -> E.z(xBz /\ E.y zAy)))
11	4, 10	19.21ai 974	. . . . 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 1016	. . . . . 6 |- (E.z(xBz /\ E.y zAy) -> E.z xBz)
14	13	ax-gen 955	. . . . 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 1083	. . . 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 1788	. . . . . 6 |- z e. V
19	18	elrn 3306	. . . . 5 |- (z e. ran B <-> E.x xBz)
20	18	eldm 3264	. . . . 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 975	. . 3 |- (A.z(z e. ran B -> z e. dom A) <-> A.z(E.x xBz -> E.y zAy))
23		visset 1788	. . . . . . 7 |- x e. V
24	23	eldm2 3265	. . . . . 6 |- (x e. dom ( A o. B) <-> E.y<.x, y>. e. (A o. B))
25		visset 1788	. . . . . . . 8 |- y e. V
26	23, 25	opelco 3245	. . . . . . 7 |- (<.x, y>. e. (A o. B) <-> E.z(xBz /\ zAy))
27	26	exbii 1027	. . . . . 6 |- (E.y<.x, y>. e. (A o. B) <-> E.yE.z(xBz /\ zAy))
28		excom 1022	. . . . . . 7 |- (E.yE.z(xBz /\ zAy) <-> E.zE.y(xBz /\ zAy))
29		19.42v 1290	. . . . . . . 8 |- (E.y(xBz /\ zAy) <-> (xBz /\ E.y zAy))
30	29	exbii 1027	. . . . . . 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 3264	. . . . 5 |- (x e. dom B <-> E.z xBz)
34	32, 33	bibi12i 608	. . . 4 |- ((x e. dom ( A o. B) <-> x e. dom B) <-> (E.z(xBz /\ E.y zAy) <-> E.z xBz))
35	34	albii 975	. . 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 2029	. 2 |- (ran B (_ dom A <-> A.z(z e. ran B -> z e. dom A))
38		dfcleq 1447	. 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 950  E.wex 956   = wceq 1099   e. wcel 1105   (_ wss 2018  <.cop 2382   class class class wbr 2587  dom cdm 3133  ran crn 3134   o. ccom 3137

This theorem is referenced by:  dmcoeq 3317  fnco 3535  fco 3575  cncfmet1 7793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-br 2588  df-opab 2635  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152

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