Theorem dfoprab3 6089

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
dfoprab3.1	|- ( w = <. x , y >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
dfoprab3	|- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }

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

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

Proof of Theorem dfoprab3

Step	Hyp	Ref	Expression
1		dfoprab3s 6088	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) }
2		vex 2689	. . . . . 6 |- w e. _V
3		1stexg 6065	. . . . . 6 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . . 5 |- ( 1st ` w ) e. _V
5		2ndexg 6066	. . . . . 6 |- ( w e. _V -> ( 2nd ` w ) e. _V )
6	2, 5	ax-mp 5	. . . . 5 |- ( 2nd ` w ) e. _V
7		eqcom 2141	. . . . . . . . . 10 |- ( x = ( 1st ` w ) <-> ( 1st ` w ) = x )
8		eqcom 2141	. . . . . . . . . 10 |- ( y = ( 2nd ` w ) <-> ( 2nd ` w ) = y )
9	7, 8	anbi12i 455	. . . . . . . . 9 |- ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) <-> ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) )
10		eqopi 6070	. . . . . . . . 9 |- ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) ) -> w = <. x , y >. )
11	9, 10	sylan2b 285	. . . . . . . 8 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> w = <. x , y >. )
12		dfoprab3.1	. . . . . . . 8 |- ( w = <. x , y >. -> ( ph <-> ps ) )
13	11, 12	syl 14	. . . . . . 7 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> ( ph <-> ps ) )
14	13	bicomd 140	. . . . . 6 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> ( ps <-> ph ) )
15	14	ex 114	. . . . 5 |- ( w e. ( _V X. _V ) -> ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) -> ( ps <-> ph ) ) )
16	4, 6, 15	sbc2iedv 2981	. . . 4 |- ( w e. ( _V X. _V ) -> ( [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps <-> ph ) )
17	16	pm5.32i 449	. . 3 |- ( ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) <-> ( w e. ( _V X. _V ) /\ ph ) )
18	17	opabbii 3995	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) }
19	1, 18	eqtr2i 2161	1 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   [.wsbc 2909   <.cop 3530   {copab 3988   X. cxp 4537   ` cfv 5123   {coprab 5775   1stc1st 6036   2ndc2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-oprab 5778  df-1st 6038  df-2nd 6039

This theorem is referenced by:  dfoprab4  6090  df1st2  6116  df2nd2  6117