Theorem opelresg 5404

Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 14-Oct-2005.)

Assertion

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

Proof of Theorem opelresg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4403	. . 3 |- ( y = B -> <. A , y >. = <. A , B >. )
2	1	eleq1d 2686	. 2 |- ( y = B -> ( <. A , y >. e. ( C |` D ) <-> <. A , B >. e. ( C |` D ) ) )
3	1	eleq1d 2686	. . 3 |- ( y = B -> ( <. A , y >. e. C <-> <. A , B >. e. C ) )
4	3	anbi1d 741	. 2 |- ( y = B -> ( ( <. A , y >. e. C /\ A e. D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )
5		vex 3203	. . 3 |- y e. _V
6	5	opelres 5401	. 2 |- ( <. A , y >. e. ( C |` D ) <-> ( <. A , y >. e. C /\ A e. D ) )
7	2, 4, 6	vtoclbg 3267	1 |- ( B e. V -> ( <. A , B >. e. ( C |` D ) <-> ( <. A , B >. e. C /\ A e. D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   |` 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:  brresg  5405  opelresi  5408  issref  5509  setsnidel  41347