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Theorem rnsnopg 5316
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 4856 . . 3  |-  ran  { <. A ,  B >. }  =  dom  `' { <. A ,  B >. }
2 dfdm4 5030 . . . 4  |-  dom  { <. B ,  A >. }  =  ran  `' { <. B ,  A >. }
3 df-rn 4856 . . . 4  |-  ran  `' { <. B ,  A >. }  =  dom  `' `' { <. B ,  A >. }
4 cnvcnvsn 5314 . . . . 5  |-  `' `' { <. B ,  A >. }  =  `' { <. A ,  B >. }
54dmeqi 5038 . . . 4  |-  dom  `' `' { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
62, 3, 53eqtri 2436 . . 3  |-  dom  { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
71, 6eqtr4i 2435 . 2  |-  ran  { <. A ,  B >. }  =  dom  { <. B ,  A >. }
8 dmsnopg 5308 . 2  |-  ( A  e.  V  ->  dom  {
<. B ,  A >. }  =  { B }
)
97, 8syl5eq 2456 1  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {csn 3782   <.cop 3785   `'ccnv 4844   dom cdm 4845   ran crn 4846
This theorem is referenced by:  rnsnop  5317  dprdsn  15557  rnpropg  23996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856
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