Theorem dfoprab3 6249

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 6248	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) }
2		vex 2766	. . . . . 6 |- w e. _V
3		1stexg 6225	. . . . . 6 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . . 5 |- ( 1st ` w ) e. _V
5		2ndexg 6226	. . . . . 6 |- ( w e. _V -> ( 2nd ` w ) e. _V )
6	2, 5	ax-mp 5	. . . . 5 |- ( 2nd ` w ) e. _V
7		eqcom 2198	. . . . . . . . . 10 |- ( x = ( 1st ` w ) <-> ( 1st ` w ) = x )
8		eqcom 2198	. . . . . . . . . 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 6230	. . . . . . . . 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 3062	. . . 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 4100	. 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 2218	1 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ ph ) } = { <. <. x , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   _Vcvv 2763   [.wsbc 2989   <.cop 3625   {copab 4093   X. cxp 4661   ` cfv 5258   {coprab 5923   1stc1st 6196   2ndc2nd 6197

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266  df-oprab 5926  df-1st 6198  df-2nd 6199

This theorem is referenced by:  dfoprab4  6250  df1st2  6277  df2nd2  6278