Theorem dfopab2 6361

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 3051	. . . . 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 6359	. . . . . . . 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 1654	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> E. y ( z = <. x , y >. /\ ph ) )
6		nfcv 2375	. . . . . . . 8 |- F/_ y ( 1st ` z )
7		nfsbc1v 3051	. . . . . . . 8 |- F/ y [. ( 2nd ` z ) / y ]. ph
8	6, 7	nfsbc 3053	. . . . . . 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 1654	. . . 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 4794	. . . . 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 4156	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
17		df-rab 2520	. 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 1398   E.wex 1541   e. wcel 2202   {cab 2217   {crab 2515   _Vcvv 2803   [.wsbc 3032   <.cop 3676   {copab 4154   X. cxp 4729   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)