Theorem rnsnop 3436

Description: The range of a singleton of an ordered pair is the singleton of the second member.

Hypotheses

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

Assertion

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

Proof of Theorem rnsnop

Step	Hyp	Ref	Expression
1		df-rn 3179	. 2 |- ran {<.A, B>.} = dom `'{<.A, B>.}
2		cnvsn.1	. . . 4 |- A e. V
3		cnvsn.2	. . . 4 |- B e. V
4	2, 3	cnvsn 3435	. . 3 |- `'{<.A, B>.} = {<.B, A>.}
5	4	dmeqi 3301	. 2 |- dom `'{<.A, B>.} = dom {<.B, A>.}
6		dmsnop 3317	. 2 |- dom {<.B, A>.} = {B}
7	1, 5, 6	3eqtr 1491	1 |- ran {<.A, B>.} = {B}

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802  {csn 2399  <.cop 2401  `'ccnv 3159  dom cdm 3160  ran crn 3161

This theorem is referenced by:  en1 4407  ringsn 8100  ghomsn 10293  1alg 10498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-cnv 3176  df-dm 3178  df-rn 3179

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