Theorem dfco2 5353

Description: Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 19-Dec-2008.)

Assertion

Ref	Expression
dfco2	|- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem dfco2

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

Step	Hyp	Ref	Expression
1		relco 5352	. 2 |- Rel ( A o. B )
2		reliun 4973	. . 3 |- ( Rel U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) <-> A. x e. _V Rel ( ( `' B " { x } ) X. ( A " { x } ) ) )
3		relxp 4961	. . . 4 |- Rel ( ( `' B " { x } ) X. ( A " { x } ) )
4	3	a1i 11	. . 3 |- ( x e. _V -> Rel ( ( `' B " { x } ) X. ( A " { x } ) ) )
5	2, 4	mprgbir 2786	. 2 |- Rel U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
6		vex 3083	. . . 4 |- y e. _V
7		vex 3083	. . . 4 |- z e. _V
8		opelco2g 5021	. . . 4 |- ( ( y e. _V /\ z e. _V ) -> ( <. y , z >. e. ( A o. B ) <-> E. x ( <. y , x >. e. B /\ <. x , z >. e. A ) ) )
9	6, 7, 8	mp2an 676	. . 3 |- ( <. y , z >. e. ( A o. B ) <-> E. x ( <. y , x >. e. B /\ <. x , z >. e. A ) )
10		eliun 4304	. . . 4 |- ( <. y , z >. e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x e. _V <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
11		rexv 3096	. . . 4 |- ( E. x e. _V <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) )
12		opelxp 4883	. . . . . 6 |- ( <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> ( y e. ( `' B " { x } ) /\ z e. ( A " { x } ) ) )
13		vex 3083	. . . . . . . . 9 |- x e. _V
14	13, 6	elimasn 5212	. . . . . . . 8 |- ( y e. ( `' B " { x } ) <-> <. x , y >. e. `' B )
15	13, 6	opelcnv 5035	. . . . . . . 8 |- ( <. x , y >. e. `' B <-> <. y , x >. e. B )
16	14, 15	bitri 252	. . . . . . 7 |- ( y e. ( `' B " { x } ) <-> <. y , x >. e. B )
17	13, 7	elimasn 5212	. . . . . . 7 |- ( z e. ( A " { x } ) <-> <. x , z >. e. A )
18	16, 17	anbi12i 701	. . . . . 6 |- ( ( y e. ( `' B " { x } ) /\ z e. ( A " { x } ) ) <-> ( <. y , x >. e. B /\ <. x , z >. e. A ) )
19	12, 18	bitri 252	. . . . 5 |- ( <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> ( <. y , x >. e. B /\ <. x , z >. e. A ) )
20	19	exbii 1712	. . . 4 |- ( E. x <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> E. x ( <. y , x >. e. B /\ <. x , z >. e. A ) )
21	10, 11, 20	3bitrri 275	. . 3 |- ( E. x ( <. y , x >. e. B /\ <. x , z >. e. A ) <-> <. y , z >. e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) )
22	9, 21	bitri 252	. 2 |- ( <. y , z >. e. ( A o. B ) <-> <. y , z >. e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) )
23	1, 5, 22	eqrelriiv 4948	1 |- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   e. wcel 1872   E.wrex 2772   _Vcvv 3080   {csn 3998   <.cop 4004   U_ciun 4299   X. cxp 4851   `'ccnv 4852   "cima 4856   o. ccom 4857   Rel wrel 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-iun 4301  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866

This theorem is referenced by:  dfco2a  5354