Theorem dfoprab2 5969

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 1678	. . . 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 1705	. . . . 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 3808	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
4	3	eqeq2d 2208	. . . . . . . . . . 11 |- ( w = <. x , y >. -> ( v = <. w , z >. <-> v = <. <. x , y >. , z >. ) )
5	4	pm5.32ri 455	. . . . . . . . . 10 |- ( ( v = <. w , z >. /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) )
6	5	anbi1i 458	. . . . . . . . 9 |- ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) )
7		anass 401	. . . . . . . . 9 |- ( ( ( v = <. w , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) )
8		an32 562	. . . . . . . . 9 |- ( ( ( v = <. <. x , y >. , z >. /\ w = <. x , y >. ) /\ ph ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
9	6, 7, 8	3bitr3i 210	. . . . . . . 8 |- ( ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
10	9	exbii 1619	. . . . . . 7 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
11		vex 2766	. . . . . . . . . 10 |- x e. _V
12		vex 2766	. . . . . . . . . 10 |- y e. _V
13	11, 12	opex 4262	. . . . . . . . 9 |- <. x , y >. e. _V
14	13	isseti 2771	. . . . . . . 8 |- E. w w = <. x , y >.
15		19.42v 1921	. . . . . . . 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 943	. . . . . . 7 |- ( E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
17	10, 16	bitri 184	. . . . . 6 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> ( v = <. <. x , y >. , z >. /\ ph ) )
18	17	3exbii 1621	. . . . 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 184	. . . 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 1926	. . . . 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 1620	. . . 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 210	. . 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 2312	. 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 5926	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
25		df-opab 4095	. 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 2227	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   {cab 2182   <.cop 3625   {copab 4093   {coprab 5923

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-oprab 5926

This theorem is referenced by:  reloprab  5970  cbvoprab1  5994  cbvoprab12  5996  cbvoprab3  5998  dmoprab  6003  rnoprab  6005  ssoprab2i  6011  mpomptx  6013  resoprab  6018  funoprabg  6021  ov6g  6061  dfoprab3s  6248  xpcomco  6885