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Theorem rnsnopg 4849
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 4402 . . 3  |-  ran  { <. A ,  B >. }  =  dom  `' { <. A ,  B >. }
2 dfdm4 4575 . . . 4  |-  dom  { <. B ,  A >. }  =  ran  `' { <. B ,  A >. }
3 df-rn 4402 . . . 4  |-  ran  `' { <. B ,  A >. }  =  dom  `' `' { <. B ,  A >. }
4 cnvcnvsn 4847 . . . . 5  |-  `' `' { <. B ,  A >. }  =  `' { <. A ,  B >. }
54dmeqi 4584 . . . 4  |-  dom  `' `' { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
62, 3, 53eqtri 2107 . . 3  |-  dom  { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
71, 6eqtr4i 2106 . 2  |-  ran  { <. A ,  B >. }  =  dom  { <. B ,  A >. }
8 dmsnopg 4842 . 2  |-  ( A  e.  V  ->  dom  {
<. B ,  A >. }  =  { B }
)
97, 8syl5eq 2127 1  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {csn 3416   <.cop 3419   `'ccnv 4390   dom cdm 4391   ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  rnpropg  4850  rnsnop  4851  fprg  5398
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