Theorem dfoprab3s 6193

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 5924	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2		nfsbc1v 2983	. . . . 5 |- F/ x [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
3	2	19.41 1686	. . . 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 6190	. . . . . . . 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 1605	. . . . . 6 |- ( E. y ( w = <. x , y >. /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) <-> E. y ( w = <. x , y >. /\ ph ) )
7		nfcv 2319	. . . . . . . 8 |- F/_ y ( 1st ` w )
8		nfsbc1v 2983	. . . . . . . 8 |- F/ y [. ( 2nd ` w ) / y ]. ph
9	7, 8	nfsbc 2985	. . . . . . 7 |- F/ y [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph
10	9	19.41 1686	. . . . . 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 1605	. . . 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 4690	. . . . 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 4072	. 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 2198	1 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | ( w e. ( _V X. _V ) /\ [. ( 1st ` w ) / x ]. [. ( 2nd ` w ) / y ]. ph ) }

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   [.wsbc 2964   <.cop 3597   {copab 4065   X. cxp 4626   ` cfv 5218   {coprab 5878   1stc1st 6141   2ndc2nd 6142

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fv 5226  df-oprab 5881  df-1st 6143  df-2nd 6144

This theorem is referenced by:  dfoprab3  6194