Theorem 2ndrn 4100

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 4098	. . 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 1546	. 2 |- ((Rel R /\ A e. R) -> <.(1st` A), (2nd` A)>. e. R)
4		fvex 3723	. . 3 |- (1st` A) e. V
5		fvex 3723	. . 3 |- (2nd` A) e. V
6	4, 5	opelrn 3340	. 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 956  <.cop 2407  ran crn 3166  Rel wrel 3170  ` cfv 3177  1stc1st 4067  2ndc2nd 4068

This theorem is referenced by:  11st22nd 10390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-1st 4069  df-2nd 4070

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