Theorem opelresALTV 34031

Description: Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Assertion

Ref	Expression
opelresALTV	|- ( C e. V -> ( <. B , C >. e. ( R |` A ) <-> ( B e. A /\ <. B , C >. e. R ) ) )

Proof of Theorem opelresALTV

Step	Hyp	Ref	Expression
1		df-res 5126	. . 3 |- ( R |` A ) = ( R i^i ( A X. _V ) )
2	1	elin2 3801	. 2 |- ( <. B , C >. e. ( R |` A ) <-> ( <. B , C >. e. R /\ <. B , C >. e. ( A X. _V ) ) )
3		elex 3212	. . . . 5 |- ( C e. V -> C e. _V )
4	3	biantrud 528	. . . 4 |- ( C e. V -> ( B e. A <-> ( B e. A /\ C e. _V ) ) )
5		opelxp 5146	. . . 4 |- ( <. B , C >. e. ( A X. _V ) <-> ( B e. A /\ C e. _V ) )
6	4, 5	syl6rbbr 279	. . 3 |- ( C e. V -> ( <. B , C >. e. ( A X. _V ) <-> B e. A ) )
7	6	anbi1cd 33997	. 2 |- ( C e. V -> ( ( <. B , C >. e. R /\ <. B , C >. e. ( A X. _V ) ) <-> ( B e. A /\ <. B , C >. e. R ) ) )
8	2, 7	syl5bb 272	1 |- ( C e. V -> ( <. B , C >. e. ( R |` A ) <-> ( B e. A /\ <. B , C >. e. R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   |` cres 5116

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-res 5126

This theorem is referenced by:  brresALTV  34032