Theorem dfoprab2 5992

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 1687	. . . 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 1714	. . . . 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 3819	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
4	3	eqeq2d 2217	. . . . . . . . . . 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 1628	. . . . . . 7 |- ( E. w ( v = <. w , z >. /\ ( w = <. x , y >. /\ ph ) ) <-> E. w ( ( v = <. <. x , y >. , z >. /\ ph ) /\ w = <. x , y >. ) )
11		vex 2775	. . . . . . . . . 10 |- x e. _V
12		vex 2775	. . . . . . . . . 10 |- y e. _V
13	11, 12	opex 4273	. . . . . . . . 9 |- <. x , y >. e. _V
14	13	isseti 2780	. . . . . . . 8 |- E. w w = <. x , y >.
15		19.42v 1930	. . . . . . . 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 944	. . . . . . 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 1630	. . . . 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 1935	. . . . 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 1629	. . . 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 2321	. 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 5948	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
25		df-opab 4106	. 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 2236	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }

Syntax hints:   /\ wa 104   = wceq 1373   E.wex 1515   {cab 2191   <.cop 3636   {copab 4104   {coprab 5945

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948

This theorem is referenced by:  reloprab  5993  cbvoprab1  6017  cbvoprab12  6019  cbvoprab3  6021  dmoprab  6026  rnoprab  6028  ssoprab2i  6034  mpomptx  6036  resoprab  6041  funoprabg  6044  ov6g  6084  dfoprab3s  6276  xpcomco  6921