Theorem rnco 5198

Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)

Assertion

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

Proof of Theorem rnco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2776	. . . . . 6 |- x e. _V
2		vex 2776	. . . . . 6 |- y e. _V
3	1, 2	brco 4857	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
4	3	exbii 1629	. . . 4 |- ( E. x x ( A o. B ) y <-> E. x E. z ( x B z /\ z A y ) )
5		excom 1688	. . . 4 |- ( E. x E. z ( x B z /\ z A y ) <-> E. z E. x ( x B z /\ z A y ) )
6		ancom 266	. . . . . . 7 |- ( ( E. x x B z /\ z A y ) <-> ( z A y /\ E. x x B z ) )
7		19.41v 1927	. . . . . . 7 |- ( E. x ( x B z /\ z A y ) <-> ( E. x x B z /\ z A y ) )
8		vex 2776	. . . . . . . . 9 |- z e. _V
9	8	elrn 4930	. . . . . . . 8 |- ( z e. ran B <-> E. x x B z )
10	9	anbi2i 457	. . . . . . 7 |- ( ( z A y /\ z e. ran B ) <-> ( z A y /\ E. x x B z ) )
11	6, 7, 10	3bitr4i 212	. . . . . 6 |- ( E. x ( x B z /\ z A y ) <-> ( z A y /\ z e. ran B ) )
12	2	brres 4974	. . . . . 6 |- ( z ( A |` ran B ) y <-> ( z A y /\ z e. ran B ) )
13	11, 12	bitr4i 187	. . . . 5 |- ( E. x ( x B z /\ z A y ) <-> z ( A |` ran B ) y )
14	13	exbii 1629	. . . 4 |- ( E. z E. x ( x B z /\ z A y ) <-> E. z z ( A |` ran B ) y )
15	4, 5, 14	3bitri 206	. . 3 |- ( E. x x ( A o. B ) y <-> E. z z ( A |` ran B ) y )
16	2	elrn 4930	. . 3 |- ( y e. ran ( A o. B ) <-> E. x x ( A o. B ) y )
17	2	elrn 4930	. . 3 |- ( y e. ran ( A |` ran B ) <-> E. z z ( A |` ran B ) y )
18	15, 16, 17	3bitr4i 212	. 2 |- ( y e. ran ( A o. B ) <-> y e. ran ( A |` ran B ) )
19	18	eqriv 2203	1 |- ran ( A o. B ) = ran ( A |` ran B )

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2177   class class class wbr 4051   ran crn 4684   |` cres 4685   o. ccom 4687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695

This theorem is referenced by:  rnco2  5199  cofunexg  6207  1stcof  6262  2ndcof  6263  djudom  7210