Theorem relopabiALT 5246

Description: Alternate proof of relopabi 5245. (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
relopabi.1	|- A = { <. x , y >. | ph }

Assertion

Ref	Expression
relopabiALT	|- Rel A

Proof of Theorem relopabiALT

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabi.1	. . . 4 |- A = { <. x , y >. | ph }
2		df-opab 4713	. . . 4 |- { <. x , y >. | ph } = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
3	1, 2	eqtri 2644	. . 3 |- A = { z | E. x E. y ( z = <. x , y >. /\ ph ) }
4		vex 3203	. . . . . . . 8 |- x e. _V
5		vex 3203	. . . . . . . 8 |- y e. _V
6	4, 5	opelvv 5166	. . . . . . 7 |- <. x , y >. e. ( _V X. _V )
7		eleq1 2689	. . . . . . 7 |- ( z = <. x , y >. -> ( z e. ( _V X. _V ) <-> <. x , y >. e. ( _V X. _V ) ) )
8	6, 7	mpbiri 248	. . . . . 6 |- ( z = <. x , y >. -> z e. ( _V X. _V ) )
9	8	adantr 481	. . . . 5 |- ( ( z = <. x , y >. /\ ph ) -> z e. ( _V X. _V ) )
10	9	exlimivv 1860	. . . 4 |- ( E. x E. y ( z = <. x , y >. /\ ph ) -> z e. ( _V X. _V ) )
11	10	abssi 3677	. . 3 |- { z | E. x E. y ( z = <. x , y >. /\ ph ) } C_ ( _V X. _V )
12	3, 11	eqsstri 3635	. 2 |- A C_ ( _V X. _V )
13		df-rel 5121	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
14	12, 13	mpbir 221	1 |- Rel A

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   <.cop 4183   {copab 4712   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)