Theorem opres 3389

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class.

Hypothesis

Ref	Expression
opres.1	|- B e. V

Assertion

Ref	Expression
opres	|- (A e. D -> (<.A, B>. e. (C |` D) <-> <.A, B>. e. C))

Proof of Theorem opres

Step	Hyp	Ref	Expression
1		opres.1	. . . 4 |- B e. V
2	1	opelres 3386	. . 3 |- (<.A, B>. e. (C |` D) <-> (<.A, B>. e. C /\ A e. D))
3	2	pm3.26bi 322	. 2 |- (<.A, B>. e. (C |` D) -> <.A, B>. e. C)
4	2	biimpr 152	. . 3 |- ((<.A, B>. e. C /\ A e. D) -> <.A, B>. e. (C |` D))
5	4	expcom 374	. 2 |- (A e. D -> (<.A, B>. e. C -> <.A, B>. e. (C |` D)))
6	3, 5	impbid2 521	1 |- (A e. D -> (<.A, B>. e. (C |` D) <-> <.A, B>. e. C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 962  Vcvv 1818  <.cop 2421   |` cres 3186

This theorem is referenced by:  resieq 3390  2elresin 3612

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-opab 2680  df-xp 3198  df-res 3204

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