Theorem dfoprab2 5889

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (Contributed by NM, 12-Mar-1995.)

Assertion

Ref	Expression
dfoprab2	|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Distinct variable groups:   x, z, w   y, z, w   ph, w

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem dfoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 1652	. . . 4 |- ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w E. z E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
2		exrot4 1679	. . . . 5 |- ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
3		opeq1 3758	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
4	3	eqeq2d 2177	. . . . . . . . . . 11 |- ( w = <. x , y >. -> ( v = <. w , z >. <-> v = <. <. x , y >. , z >. ) )
5	4	pm5.32ri 451	. . . . . . . . . 10 |- ( ( v = <. w , z >. /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) )
6	5	anbi1i 454	. . . . . . . . 9 |- ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) )
7		anass 399	. . . . . . . . 9 |- ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
8		an32 552	. . . . . . . . 9 |- ( ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
9	6, 7, 8	3bitr3i 209	. . . . . . . 8 |- ( ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
10	9	exbii 1593	. . . . . . 7 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
11		vex 2729	. . . . . . . . . 10 |- x e. _V
12		vex 2729	. . . . . . . . . 10 |- y e. _V
13	11, 12	opex 4207	. . . . . . . . 9 |- <. x , y >. e. _V
14	13	isseti 2734	. . . . . . . 8 |- E. w w = <. x , y >.
15		19.42v 1894	. . . . . . . 8 |- ( E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ E. w w = <. x , y >. ) )
16	14, 15	mpbiran2 931	. . . . . . 7 |- ( E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
17	10, 16	bitri 183	. . . . . 6 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
18	17	3exbii 1595	. . . . 5 |- ( E. x E. y E. z E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) )
19	2, 18	bitri 183	. . . 4 |- ( E. z E. w E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) )
20		19.42vv 1899	. . . . 5 |- ( E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
21	20	2exbii 1594	. . . 4 |- ( E. w E. z E. x E. y ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
22	1, 19, 21	3bitr3i 209	. . 3 |- ( E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) )
23	22	abbii 2282	. 2 |- { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) } = { v | E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
24		df-oprab 5846	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
25		df-opab 4044	. 2 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { v | E. w E. z ( v = <. w , z >. /\ E. x E. y ( w = <. x , y >. /\ ph ) ) }
26	23, 24, 25	3eqtr4i 2196	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   {cab 2151   <.cop 3579   {copab 4042   {coprab 5843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-oprab 5846

This theorem is referenced by:  reloprab  5890  cbvoprab1  5914  cbvoprab12  5916  cbvoprab3  5918  dmoprab  5923  rnoprab  5925  ssoprab2i  5931  mpomptx  5933  resoprab  5938  funoprabg  5941  ov6g  5979  dfoprab3s  6158  xpcomco  6792