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Theorem rnsnopg 5109
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 4639 . . 3  |-  ran  { <. A ,  B >. }  =  dom  `' { <. A ,  B >. }
2 dfdm4 4821 . . . 4  |-  dom  { <. B ,  A >. }  =  ran  `' { <. B ,  A >. }
3 df-rn 4639 . . . 4  |-  ran  `' { <. B ,  A >. }  =  dom  `' `' { <. B ,  A >. }
4 cnvcnvsn 5107 . . . . 5  |-  `' `' { <. B ,  A >. }  =  `' { <. A ,  B >. }
54dmeqi 4830 . . . 4  |-  dom  `' `' { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
62, 3, 53eqtri 2202 . . 3  |-  dom  { <. B ,  A >. }  =  dom  `' { <. A ,  B >. }
71, 6eqtr4i 2201 . 2  |-  ran  { <. A ,  B >. }  =  dom  { <. B ,  A >. }
8 dmsnopg 5102 . 2  |-  ( A  e.  V  ->  dom  {
<. B ,  A >. }  =  { B }
)
97, 8eqtrid 2222 1  |-  ( A  e.  V  ->  ran  {
<. A ,  B >. }  =  { B }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   {csn 3594   <.cop 3597   `'ccnv 4627   dom cdm 4628   ran crn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639
This theorem is referenced by:  rnpropg  5110  rnsnop  5111  fprg  5701
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