Theorem dfoprab3 6187

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 6186	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) }
2		vex 2740	. . . . . 6 |- w e. _V
3		1stexg 6163	. . . . . 6 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . . 5 |- ( 1st ` w ) e. _V
5		2ndexg 6164	. . . . . 6 |- ( w e. _V -> ( 2nd ` w ) e. _V )
6	2, 5	ax-mp 5	. . . . 5 |- ( 2nd ` w ) e. _V
7		eqcom 2179	. . . . . . . . . 10 |- ( x = ( 1st ` w ) <-> ( 1st ` w ) = x )
8		eqcom 2179	. . . . . . . . . 10 |- ( y = ( 2nd ` w ) <-> ( 2nd ` w ) = y )
9	7, 8	anbi12i 460	. . . . . . . . 9 |- ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) <-> ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) )
10		eqopi 6168	. . . . . . . . 9 |- ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) ) -> w = <. x , y >. )
11	9, 10	sylan2b 287	. . . . . . . 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 141	. . . . . 6 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> ( ps <-> ph ) )
15	14	ex 115	. . . . 5 |- ( w e. ( _V X. _V ) -> ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) -> ( ps <-> ph ) ) )
16	4, 6, 15	sbc2iedv 3035	. . . 4 |- ( w e. ( _V X. _V ) -> ( [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps <-> ph ) )
17	16	pm5.32i 454	. . 3 |- ( ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) <-> ( w e. ( _V X. _V ) /\ ph ) )
18	17	opabbii 4068	. 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 2199	1 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   [.wsbc 2962   <.cop 3595   {copab 4061   X. cxp 4622   ` cfv 5213   {coprab 5871   1stc1st 6134   2ndc2nd 6135

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-fo 5219  df-fv 5221  df-oprab 5874  df-1st 6136  df-2nd 6137

This theorem is referenced by:  dfoprab4  6188  df1st2  6215  df2nd2  6216