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Theorem rnsnopg 5152
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 4700 . . 3  |-  ran  { <. A ,  B >. }  =  dom  `' { <. A ,  B >. }
2 dfdm4 4872 . . . 4  |-  dom  { <. B ,  A >. }  =  ran  `' { <. B ,  A >. }
3 df-rn 4700 . . . 4  |-  ran  `' { <. B ,  A >. }  =  dom  `' `' { <. B ,  A >. }
4 cnvcnvsn 5150 . . . . 5  |-  `' `' { <. B ,  A >. }  =  `' { <. A ,  B >. }
54dmeqi 4880 . . . 4  |-  dom  `' `' { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
62, 3, 53eqtri 2307 . . 3  |-  dom  { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
71, 6eqtr4i 2306 . 2  |-  ran  { <. A ,  B >. }  =  dom  { <. B ,  A >. }
8 dmsnopg 5144 . 2  |-  ( A  e.  V  ->  dom  {
<. B ,  A >. }  =  { B }
)
97, 8syl5eq 2327 1  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  rnsnop  5153  dprdsn  15271  rnpropg  23189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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