Theorem 2ndrn 4110

Description: The second ordered pair component of a member of a relation belongs to the range of the relation.

Assertion

Ref	Expression
2ndrn	|- ((Rel R /\ A e. R) -> (2nd` A) e. ran R)

Proof of Theorem 2ndrn

Step	Hyp	Ref	Expression
1		1st2nd 4108	. . 3 |- ((Rel R /\ A e. R) -> A = <.(1st` A), (2nd` A)>.)
2		pm3.27 323	. . 3 |- ((Rel R /\ A e. R) -> A e. R)
3	1, 2	eqeltrrd 1549	. 2 |- ((Rel R /\ A e. R) -> <.(1st` A), (2nd` A)>. e. R)
4		fvex 3732	. . 3 |- (1st` A) e. V
5		fvex 3732	. . 3 |- (2nd` A) e. V
6	4, 5	opelrn 3345	. 2 |- (<.(1st` A), (2nd` A)>. e. R -> (2nd` A) e. ran R)
7	3, 6	syl 10	1 |- ((Rel R /\ A e. R) -> (2nd` A) e. ran R)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  <.cop 2411  ran crn 3171  Rel wrel 3175  ` cfv 3182  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  11st22nd 10458

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-9 965  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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-ral 1649  df-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080

Copyright terms: Public domain