Theorem dfoprab3s 6158

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 5889	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfsbc1v 2969	. . . . 5 |- F/ x [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
3	2	19.41 1674	. . . 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 6155	. . . . . . . 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 1593	. . . . . 6 |- ( E. y ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> E. y ( w = <. x , y >. /\ ph ) )
7		nfcv 2308	. . . . . . . 8 |- F/_ y ( 1st ` w )
8		nfsbc1v 2969	. . . . . . . 8 |- F/ y [. ( 2nd ` w ) / y ]. ph
9	7, 8	nfsbc 2971	. . . . . . 7 |- F/ y [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
10	9	19.41 1674	. . . . . 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 1593	. . . 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 4666	. . . . 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 4049	. 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 2186	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   [.wsbc 2951   <.cop 3579   {copab 4042   X. cxp 4602   ` cfv 5188   {coprab 5843   1stc1st 6106   2ndc2nd 6107

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-oprab 5846  df-1st 6108  df-2nd 6109

This theorem is referenced by:  dfoprab3  6159