Theorem opelrn 3345

Description: Membership of second member of an ordered pair in a range.

Hypotheses

Ref	Expression
brelrn.1	|- A e. V
brelrn.2	|- B e. V

Assertion

Ref	Expression
opelrn	|- (<.A, B>. e. C -> B e. ran C)

Proof of Theorem opelrn

Step	Hyp	Ref	Expression
1		df-br 2620	. 2 |- (ACB <-> <.A, B>. e. C)
2		brelrn.1	. . 3 |- A e. V
3		brelrn.2	. . 3 |- B e. V
4	2, 3	brelrn 3344	. 2 |- (ACB -> B e. ran C)
5	1, 4	sylbir 201	1 |- (<.A, B>. e. C -> B e. ran C)

Syntax hints:   -> wi 3   e. wcel 958  Vcvv 1811  <.cop 2411   class class class wbr 2619  ran crn 3171

This theorem is referenced by:  zfrep6 3614  2ndrn 4110

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

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