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Theorem rnsnopg 5352
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, 30-Apr-2015.)
Assertion
Ref Expression
rnsnopg  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)

Proof of Theorem rnsnopg
StepHypRef Expression
1 df-rn 4892 . . 3  |-  ran  { <. A ,  B >. }  =  dom  `' { <. A ,  B >. }
2 dfdm4 5066 . . . 4  |-  dom  { <. B ,  A >. }  =  ran  `' { <. B ,  A >. }
3 df-rn 4892 . . . 4  |-  ran  `' { <. B ,  A >. }  =  dom  `' `' { <. B ,  A >. }
4 cnvcnvsn 5350 . . . . 5  |-  `' `' { <. B ,  A >. }  =  `' { <. A ,  B >. }
54dmeqi 5074 . . . 4  |-  dom  `' `' { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
62, 3, 53eqtri 2462 . . 3  |-  dom  { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
71, 6eqtr4i 2461 . 2  |-  ran  { <. A ,  B >. }  =  dom  { <. B ,  A >. }
8 dmsnopg 5344 . 2  |-  ( A  e.  V  ->  dom  {
<. B ,  A >. }  =  { B }
)
97, 8syl5eq 2482 1  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {csn 3816   <.cop 3819   `'ccnv 4880   dom cdm 4881   ran crn 4882
This theorem is referenced by:  rnsnop  5353  dprdsn  15599  rnpropg  24040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892
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