Theorem dfopab2 6351

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

Assertion

Ref	Expression
dfopab2	|- { <. x , y >. | ph } = { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph }

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

Allowed substitution hints:   ph( x, y)

Proof of Theorem dfopab2

Step	Hyp	Ref	Expression
1		nfsbc1v 3050	. . . . 5 |- F/ x [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph
2	1	19.41 1734	. . . 4 |- ( E. x ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. x E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
3		sbcopeq1a 6349	. . . . . . . 8 |- ( z = <. x , y >. -> ( [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph <-> ph ) )
4	3	pm5.32i 454	. . . . . . 7 |- ( ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( z = <. x , y >. /\ ph ) )
5	4	exbii 1653	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> E. y ( z = <. x , y >. /\ ph ) )
6		nfcv 2374	. . . . . . . 8 |- F/_ y ( 1st ` z )
7		nfsbc1v 3050	. . . . . . . 8 |- F/ y [. ( 2nd ` z ) / y ]. ph
8	6, 7	nfsbc 3052	. . . . . . 7 |- F/ y [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph
9	8	19.41 1734	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
10	5, 9	bitr3i 186	. . . . 5 |- ( E. y ( z = <. x , y >. /\ ph ) <-> ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
11	10	exbii 1653	. . . 4 |- ( E. x E. y ( z = <. x , y >. /\ ph ) <-> E. x ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
12		elvv 4788	. . . . 5 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
13	12	anbi1i 458	. . . 4 |- ( ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( E. x E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
14	2, 11, 13	3bitr4i 212	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
15	14	abbii 2347	. 2 |- { z | E. x E. y ( z = <. x , y >. /\ ph ) } = { z | ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) }
16		df-opab 4151	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
17		df-rab 2519	. 2 |- { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph } = { z | ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) }
18	15, 16, 17	3eqtr4i 2262	1 |- { <. x , y >. | ph } = { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph }

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   [.wsbc 3031   <.cop 3672   {copab 4149   X. cxp 4723   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by: (None)