Theorem dfoprab3s 6128

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 5858	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfsbc1v 2951	. . . . 5 |- F/ x [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
3	2	19.41 1663	. . . 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 6125	. . . . . . . 8 |- ( w = <. x , y >. -> ( [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph <-> ph ) )
5	4	pm5.32i 450	. . . . . . 7 |- ( ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> ( w = <. x , y >. /\ ph ) )
6	5	exbii 1582	. . . . . 6 |- ( E. y ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> E. y ( w = <. x , y >. /\ ph ) )
7		nfcv 2296	. . . . . . . 8 |- F/_ y ( 1st ` w )
8		nfsbc1v 2951	. . . . . . . 8 |- F/ y [. ( 2nd ` w ) / y ]. ph
9	7, 8	nfsbc 2953	. . . . . . 7 |- F/ y [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
10	9	19.41 1663	. . . . . 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 185	. . . . 5 |- ( E. y ( w = <. x , y >. /\ ph ) <-> ( E. y w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
12	11	exbii 1582	. . . 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 4641	. . . . 5 |- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. )
14	13	anbi1i 454	. . . 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 211	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) )
16	15	opabbii 4027	. 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 2175	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }

Syntax hints:   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 2125   _Vcvv 2709   [.wsbc 2933   <.cop 3559   {copab 4020   X. cxp 4577   ` cfv 5163   {coprab 5815   1stc1st 6076   2ndc2nd 6077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fv 5171  df-oprab 5818  df-1st 6078  df-2nd 6079

This theorem is referenced by:  dfoprab3  6129