Theorem rnsnop 5019

Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
cnvsn.1	|- A e. _V

Assertion

Ref	Expression
rnsnop	|- ran { <. A , B >. } = { B }

Proof of Theorem rnsnop

Step	Hyp	Ref	Expression
1		cnvsn.1	. 2 |- A e. _V
2		rnsnopg 5017	. 2 |- ( A e. _V -> ran { <. A , B >. } = { B } )
3	1, 2	ax-mp 5	1 |- ran { <. A , B >. } = { B }

Syntax hints:   = wceq 1331   e. wcel 1480   _Vcvv 2686   {csn 3527   <.cop 3530   ran crn 4540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550

This theorem is referenced by:  op2nda  5023  fpr  5602  en1  6693