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Theorem rnpropg 5126
Description: The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Assertion
Ref Expression
rnpropg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
)

Proof of Theorem rnpropg
StepHypRef Expression
1 df-pr 3614 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
21rneqi 4873 . 2  |-  ran  { <. A ,  C >. , 
<. B ,  D >. }  =  ran  ( {
<. A ,  C >. }  u.  { <. B ,  D >. } )
3 rnsnopg 5125 . . . . 5  |-  ( A  e.  V  ->  ran  {
<. A ,  C >. }  =  { C }
)
43adantr 276 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. }  =  { C } )
5 rnsnopg 5125 . . . . 5  |-  ( B  e.  W  ->  ran  {
<. B ,  D >. }  =  { D }
)
65adantl 277 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. B ,  D >. }  =  { D } )
74, 6uneq12d 3305 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )  =  ( { C }  u.  { D } ) )
8 rnun 5055 . . 3  |-  ran  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )  =  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )
9 df-pr 3614 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
107, 8, 93eqtr4g 2247 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } )  =  { C ,  D } )
112, 10eqtrid 2234 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. ,  <. B ,  D >. }  =  { C ,  D }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    u. cun 3142   {csn 3607   {cpr 3608   <.cop 3610   ran crn 4645
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655
This theorem is referenced by: (None)
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