Theorem rnco 3508

Description: The range of the composition of two classes.

Assertion

Ref	Expression
rnco	|- ran ( A o. B) = ran ( A |` ran B)

Proof of Theorem rnco

Step	Hyp	Ref	Expression
1		visset 1816	. . . . . 6 |- x e. V
2		visset 1816	. . . . . 6 |- y e. V
3	1, 2	brco 3295	. . . . 5 |- (x(A o. B)y <-> E.z(xBz /\ zAy))
4	3	exbii 1053	. . . 4 |- (E.x x(A o. B)y <-> E.xE.z(xBz /\ zAy))
5		excom 1048	. . . 4 |- (E.xE.z(xBz /\ zAy) <-> E.zE.x(xBz /\ zAy))
6		ancom 437	. . . . . . 7 |- ((E.x xBz /\ zAy) <-> (zAy /\ E.x xBz))
7		19.41v 1307	. . . . . . 7 |- (E.x(xBz /\ zAy) <-> (E.x xBz /\ zAy))
8		visset 1816	. . . . . . . . 9 |- z e. V
9	8	elrn 3356	. . . . . . . 8 |- (z e. ran B <-> E.x xBz)
10	9	anbi2i 482	. . . . . . 7 |- ((zAy /\ z e. ran B) <-> (zAy /\ E.x xBz))
11	6, 7, 10	3bitr4 183	. . . . . 6 |- (E.x(xBz /\ zAy) <-> (zAy /\ z e. ran B))
12	2	brres 3379	. . . . . 6 |- (z(A |` ran B)y <-> (zAy /\ z e. ran B))
13	11, 12	bitr4 176	. . . . 5 |- (E.x(xBz /\ zAy) <-> z(A |` ran B)y)
14	13	exbii 1053	. . . 4 |- (E.zE.x(xBz /\ zAy) <-> E.z z(A |` ran B)y)
15	4, 5, 14	3bitr 177	. . 3 |- (E.x x(A o. B)y <-> E.z z(A |` ran B)y)
16	2	elrn 3356	. . 3 |- (y e. ran ( A o. B) <-> E.x x(A o. B)y)
17	2	elrn 3356	. . 3 |- (y e. ran ( A |` ran B) <-> E.z z(A |` ran B)y)
18	15, 16, 17	3bitr4 183	. 2 |- (y e. ran ( A o. B) <-> y e. ran ( A |` ran B))
19	18	eqriv 1477	1 |- ran ( A o. B) = ran ( A |` ran B)

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   class class class wbr 2624  ran crn 3177   |` cres 3178   o. ccom 3180

This theorem is referenced by:  rnco2 3509  cofunexg 3586  1stcof 4107

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196

Copyright terms: Public domain