Theorem dfoprab2 5916

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 1664	. . . 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 1691	. . . . 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 3776	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
4	3	eqeq2d 2189	. . . . . . . . . . 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 1605	. . . . . . 7 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
11		vex 2740	. . . . . . . . . 10 |- x e. _V
12		vex 2740	. . . . . . . . . 10 |- y e. _V
13	11, 12	opex 4226	. . . . . . . . 9 |- <. x , y >. e. _V
14	13	isseti 2745	. . . . . . . 8 |- E. w w = <. x , y >.
15		19.42v 1906	. . . . . . . 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 941	. . . . . . 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 1607	. . . . 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 1911	. . . . 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 1606	. . . 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 2293	. 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 5873	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
25		df-opab 4062	. 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 2208	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   {cab 2163   <.cop 3594   {copab 4060   {coprab 5870

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-oprab 5873

This theorem is referenced by:  reloprab  5917  cbvoprab1  5941  cbvoprab12  5943  cbvoprab3  5945  dmoprab  5950  rnoprab  5952  ssoprab2i  5958  mpomptx  5960  resoprab  5965  funoprabg  5968  ov6g  6006  dfoprab3s  6185  xpcomco  6820