Theorem dfoprab3 6244

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 6243	. 2 |- { <. <. x , y >. , z >. | ps } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ps ) }
2		vex 2763	. . . . . 6 |- w e. _V
3		1stexg 6220	. . . . . 6 |- ( w e. _V -> ( 1st ` w ) e. _V )
4	2, 3	ax-mp 5	. . . . 5 |- ( 1st ` w ) e. _V
5		2ndexg 6221	. . . . . 6 |- ( w e. _V -> ( 2nd ` w ) e. _V )
6	2, 5	ax-mp 5	. . . . 5 |- ( 2nd ` w ) e. _V
7		eqcom 2195	. . . . . . . . . 10 |- ( x = ( 1st ` w ) <-> ( 1st ` w ) = x )
8		eqcom 2195	. . . . . . . . . 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 6225	. . . . . . . . 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 3058	. . . 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 4096	. 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 2215	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 2164   _Vcvv 2760   [.wsbc 2985   <.cop 3621   {copab 4089   X. cxp 4657   ` cfv 5254   {coprab 5919   1stc1st 6191   2ndc2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-oprab 5922  df-1st 6193  df-2nd 6194

This theorem is referenced by:  dfoprab4  6245  df1st2  6272  df2nd2  6273