Theorem dfco2 5282

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 5281	. 2 |- Rel ( A o. B )
2		reliun 4893	. . 3 |- ( Rel U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) <-> A. x e. _V Rel ( ( `' B " { x } ) X. ( A " { x } ) ) )
3		relxp 4879	. . . 4 |- Rel ( ( `' B " { x } ) X. ( A " { x } ) )
4	3	a1i 9	. . 3 |- ( x e. _V -> Rel ( ( `' B " { x } ) X. ( A " { x } ) ) )
5	2, 4	mprgbir 2608	. 2 |- Rel U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )
6		vex 2824	. . . 4 |- y e. _V
7		vex 2824	. . . 4 |- z e. _V
8		opelco2g 4943	. . . 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 430	. . 3 |- ( <. y , z >. e. ( A o. B ) <-> E. x ( <. y , x >. e. B /\ <. x , z >. e. A ) )
10		eliun 4011	. . . 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 2840	. . . 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 4799	. . . . . 6 |- ( <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> ( y e. ( `' B " { x } ) /\ z e. ( A " { x } ) ) )
13		vex 2824	. . . . . . . . 9 |- x e. _V
14	13, 6	elimasn 5149	. . . . . . . 8 |- ( y e. ( `' B " { x } ) <-> <. x , y >. e. `' B )
15	13, 6	opelcnv 4957	. . . . . . . 8 |- ( <. x , y >. e. `' B <-> <. y , x >. e. B )
16	14, 15	bitri 184	. . . . . . 7 |- ( y e. ( `' B " { x } ) <-> <. y , x >. e. B )
17	13, 7	elimasn 5149	. . . . . . 7 |- ( z e. ( A " { x } ) <-> <. x , z >. e. A )
18	16, 17	anbi12i 464	. . . . . 6 |- ( ( y e. ( `' B " { x } ) /\ z e. ( A " { x } ) ) <-> ( <. y , x >. e. B /\ <. x , z >. e. A ) )
19	12, 18	bitri 184	. . . . 5 |- ( <. y , z >. e. ( ( `' B " { x } ) X. ( A " { x } ) ) <-> ( <. y , x >. e. B /\ <. x , z >. e. A ) )
20	19	exbii 1658	. . . 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 207	. . 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 184	. 2 |- ( <. y , z >. e. ( A o. B ) <-> <. y , z >. e. U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) ) )
23	1, 5, 22	eqrelriiv 4864	1 |- ( A o. B ) = U_ x e. _V ( ( `' B " { x } ) X. ( A " { x } ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   _Vcvv 2821   {csn 3705   <.cop 3708   U_ciun 4007   X. cxp 4767   `'ccnv 4768   "cima 4772   o. ccom 4773   Rel wrel 4774

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-iun 4009  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  dfco2a  5283