Theorem dmcosseq 2572

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 709	. . . . . . . 8 |- (E.x xBz -> A.xE.x xBz)
2		ax-17 925	. . . . . . . 8 |- (E.y zAy -> A.xE.y zAy)
3	1, 2	hbim 702	. . . . . . 7 |- ((E.x xBz -> E.y zAy) -> A.x(E.x xBz -> E.y zAy))
4	3	hbal 700	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> A.xA.z(E.x xBz -> E.y zAy))
5		hba1 698	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> A.zA.z(E.x xBz -> E.y zAy))
6		19.8a 712	. . . . . . . . . 10 |- (xBz -> E.x xBz)
7	6	syl4 19	. . . . . . . . 9 |- ((E.x xBz -> E.y zAy) -> (xBz -> E.y zAy))
8	7	ancld 246	. . . . . . . 8 |- ((E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
9	8	a4s 682	. . . . . . 7 |- (A.z(E.x xBz -> E.y zAy) -> (xBz -> (xBz /\ E.y zAy)))
10	5, 9	19.22d 744	. . . . . 6 |- (A.z(E.x xBz -> E.y zAy) -> (E.z xBz -> E.z(xBz /\ E.y zAy)))
11	4, 10	19.21ai 740	. . . . 5 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z xBz -> E.z(xBz /\ E.y zAy)))
12		pm3.26 256	. . . . . . 7 |- ((xBz /\ E.y zAy) -> xBz)
13	12	19.22i 723	. . . . . 6 |- (E.z(xBz /\ E.y zAy) -> E.z xBz)
14	13	ax-gen 677	. . . . 5 |- A.x(E.z(xBz /\ E.y zAy) -> E.z xBz)
15	11, 14	jctil 240	. . . 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 785	. . . 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 175	. . 3 |- (A.z(E.x xBz -> E.y zAy) -> A.x(E.z(xBz /\ E.y zAy) <-> E.z xBz))
18		visset 1350	. . . . . 6 |- z e. V
19	18	elrn2 2563	. . . . 5 |- (z e. ran B <-> E.x xBz)
20	18	eldm 2527	. . . . 5 |- (z e. dom A <-> E.y zAy)
21	19, 20	imbi12i 163	. . . 4 |- ((z e. ran B -> z e. dom A) <-> (E.x xBz -> E.y zAy))
22	21	bial 695	. . 3 |- (A.z(z e. ran B -> z e. dom A) <-> A.z(E.x xBz -> E.y zAy))
23		visset 1350	. . . . . . 7 |- x e. V
24	23	eldm2 2528	. . . . . 6 |- (x e. dom (A o. B) <-> E.y<.x, y>. e. (A o. B))
25		visset 1350	. . . . . . . 8 |- y e. V
26	23, 25	opelco 2509	. . . . . . 7 |- (<.x, y>. e. (A o. B) <-> E.z(xBz /\ zAy))
27	26	biex 733	. . . . . 6 |- (E.y<.x, y>. e. (A o. B) <-> E.yE.z(xBz /\ zAy))
28		excom 728	. . . . . . 7 |- (E.yE.z(xBz /\ zAy) <-> E.zE.y(xBz /\ zAy))
29		19.42v 966	. . . . . . . 8 |- (E.y(xBz /\ zAy) <-> (xBz /\ E.y zAy))
30	29	biex 733	. . . . . . 7 |- (E.zE.y(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
31	28, 30	bitr 151	. . . . . 6 |- (E.yE.z(xBz /\ zAy) <-> E.z(xBz /\ E.y zAy))
32	24, 27, 31	3bitr 155	. . . . 5 |- (x e. dom (A o. B) <-> E.z(xBz /\ E.y zAy))
33	23	eldm 2527	. . . . 5 |- (x e. dom B <-> E.z xBz)
34	32, 33	bibi12i 462	. . . 4 |- ((x e. dom (A o. B) <-> x e. dom B) <-> (E.z(xBz /\ E.y zAy) <-> E.z xBz))
35	34	bial 695	. . 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 192	. 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 1497	. 2 |- (ran B (_ dom A <-> A.z(z e. ran B -> z e. dom A))
38		dfcleq 1098	. 2 |- (dom (A o. B) = dom B <-> A.x(x e. dom (A o. B) <-> x e. dom B))
39	36, 37, 38	3imtr4 192	1 |- (ran B (_ dom A -> dom (A o. B) = dom B)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092   (_ wss 1487  <.cop 1810   class class class wbr 2054  dom cdm 2410  ran crn 2411   o. ccom 2414

This theorem is referenced by:  dmcoeq 2573  fco 2760

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429

metamath.org