Theorem dfoprab3 7224

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 7223	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) }
2		fvex 6201	. . . . 5 |- ( 1st ` w ) e. _V
3		fvex 6201	. . . . 5 |- ( 2nd ` w ) e. _V
4		eqcom 2629	. . . . . . . . . 10 |- ( x = ( 1st ` w ) <-> ( 1st ` w ) = x )
5		eqcom 2629	. . . . . . . . . 10 |- ( y = ( 2nd ` w ) <-> ( 2nd ` w ) = y )
6	4, 5	anbi12i 733	. . . . . . . . 9 |- ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) <-> ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) )
7		eqopi 7202	. . . . . . . . 9 |- ( ( w e. ( _V X. _V ) /\ ( ( 1st ` w ) = x /\ ( 2nd ` w ) = y ) ) -> w = <. x , y >. )
8	6, 7	sylan2b 492	. . . . . . . 8 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> w = <. x , y >. )
9		dfoprab3.1	. . . . . . . 8 |- ( w = <. x , y >. -> ( ph <-> ps ) )
10	8, 9	syl 17	. . . . . . 7 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> ( ph <-> ps ) )
11	10	bicomd 213	. . . . . 6 |- ( ( w e. ( _V X. _V ) /\ ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) ) -> ( ps <-> ph ) )
12	11	ex 450	. . . . 5 |- ( w e. ( _V X. _V ) -> ( ( x = ( 1st ` w ) /\ y = ( 2nd ` w ) ) -> ( ps <-> ph ) ) )
13	2, 3, 12	sbc2iedv 3506	. . . 4 |- ( w e. ( _V X. _V ) -> ( [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps <-> ph ) )
14	13	pm5.32i 669	. . 3 |- ( ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) <-> ( w e. ( _V X. _V ) /\ ph ) )
15	14	opabbii 4717	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) }
16	1, 15	eqtr2i 2645	1 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   <.cop 4183   {copab 4712   X. cxp 5112   ` cfv 5888   {coprab 6651   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-1st 7168  df-2nd 7169

This theorem is referenced by:  dfoprab4  7225  df1st2  7263  df2nd2  7264