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Theorem rnpropg 4830
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 3413 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
21rneqi 4590 . 2  |-  ran  { <. A ,  C >. , 
<. B ,  D >. }  =  ran  ( {
<. A ,  C >. }  u.  { <. B ,  D >. } )
3 rnsnopg 4829 . . . . 5  |-  ( A  e.  V  ->  ran  {
<. A ,  C >. }  =  { C }
)
43adantr 270 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. }  =  { C } )
5 rnsnopg 4829 . . . . 5  |-  ( B  e.  W  ->  ran  {
<. B ,  D >. }  =  { D }
)
65adantl 271 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. B ,  D >. }  =  { D } )
74, 6uneq12d 3128 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )  =  ( { C }  u.  { D } ) )
8 rnun 4762 . . 3  |-  ran  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )  =  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )
9 df-pr 3413 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
107, 8, 93eqtr4g 2139 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } )  =  { C ,  D } )
112, 10syl5eq 2126 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 102    = wceq 1285    e. wcel 1434    u. cun 2972   {csn 3406   {cpr 3407   <.cop 3409   ran crn 4372
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by: (None)
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