Theorem 2ndrn 5050

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 5048	. . 3 |- ((Rel R /\ A e. R) -> A = <.(1st` A), (2nd` A)>.)
2		simpr 350	. . 3 |- ((Rel R /\ A e. R) -> A e. R)
3	1, 2	eqeltrrd 1972	. 2 |- ((Rel R /\ A e. R) -> <.(1st` A), (2nd` A)>. e. R)
4		fvex 4689	. . 3 |- (1st` A) e. _V
5		fvex 4689	. . 3 |- (2nd` A) e. _V
6	4, 5	opelrn 4192	. 2 |- (<.(1st` A), (2nd` A)>. e. R -> (2nd` A) e. ran R)
7	3, 6	syl 12	1 |- ((Rel R /\ A e. R) -> (2nd` A) e. ran R)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  <.cop 3046  ran crn 3987  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  11st22nd 14348  issubcat 15193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

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