Theorem dfoprab3s 6243

Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfoprab3s	|- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }

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

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

Proof of Theorem dfoprab3s

Step	Hyp	Ref	Expression
1		dfoprab2 5965	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfsbc1v 3004	. . . . 5 |- F/ x [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
3	2	19.41 1697	. . . 4 |- ( E. x ( E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> ( E. x E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
4		sbcopeq1a 6240	. . . . . . . 8 |- ( w = <. x , y >. -> ( [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph <-> ph ) )
5	4	pm5.32i 454	. . . . . . 7 |- ( ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> ( w = <. x , y >. /\ ph ) )
6	5	exbii 1616	. . . . . 6 |- ( E. y ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> E. y ( w = <. x , y >. /\ ph ) )
7		nfcv 2336	. . . . . . . 8 |- F/_ y ( 1st ` w )
8		nfsbc1v 3004	. . . . . . . 8 |- F/ y [. ( 2nd ` w ) / y ]. ph
9	7, 8	nfsbc 3006	. . . . . . 7 |- F/ y [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
10	9	19.41 1697	. . . . . 6 |- ( E. y ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> ( E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
11	6, 10	bitr3i 186	. . . . 5 |- ( E. y ( w = <. x , y >. /\ ph ) <-> ( E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
12	11	exbii 1616	. . . 4 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x ( E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
13		elvv 4721	. . . . 5 |- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. )
14	13	anbi1i 458	. . . 4 |- ( ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> ( E. x E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
15	3, 12, 14	3bitr4i 212	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
16	15	opabbii 4096	. 2 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }
17	1, 16	eqtri 2214	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   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-fv 5262  df-oprab 5922  df-1st 6193  df-2nd 6194

This theorem is referenced by:  dfoprab3  6244