Theorem dfopab2 5815

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 2782	. . . . 5 |- F/ x [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph
2	1	19.41 1576	. . . 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 5813	. . . . . . . 8 |- ( z = <. x , y >. -> ( [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph <-> ph ) )
4	3	pm5.32i 427	. . . . . . 7 |- ( ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> ( z = <. x , y >. /\ ph ) )
5	4	exbii 1496	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) <-> E. y ( z = <. x , y >. /\ ph ) )
6		nfcv 2178	. . . . . . . 8 |- F/_ y ( 1st ` z )
7		nfsbc1v 2782	. . . . . . . 8 |- F/ y [. ( 2nd ` z ) / y ]. ph
8	6, 7	nfsbc 2784	. . . . . . 7 |- F/ y [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph
9	8	19.41 1576	. . . . . 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 175	. . . . 5 |- ( E. y ( z = <. x , y >. /\ ph ) <-> ( E. y z = <. x , y >. /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
11	10	exbii 1496	. . . 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 4402	. . . . 5 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
13	12	anbi1i 431	. . . 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 201	. . 3 |- ( E. x E. y ( z = <. x , y >. /\ ph ) <-> ( z e. ( _V X. _V ) /\ [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph ) )
15	14	abbii 2153	. 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 3819	. 2 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
17		df-rab 2315	. 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 2070	1 |- { <. x , y >. | ph } = { z e. ( _V X. _V ) | [. ( 1st ` z ) / x ]. [. ( 2nd ` z ) / y ]. ph }

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   {crab 2310   _Vcvv 2557   [.wsbc 2764   <.cop 3378   {copab 3817   X. cxp 4343   ` cfv 4902   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by: (None)