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Theorem rnpropg 4988
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 3504 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
21rneqi 4737 . 2  |-  ran  { <. A ,  C >. , 
<. B ,  D >. }  =  ran  ( {
<. A ,  C >. }  u.  { <. B ,  D >. } )
3 rnsnopg 4987 . . . . 5  |-  ( A  e.  V  ->  ran  {
<. A ,  C >. }  =  { C }
)
43adantr 274 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. }  =  { C } )
5 rnsnopg 4987 . . . . 5  |-  ( B  e.  W  ->  ran  {
<. B ,  D >. }  =  { D }
)
65adantl 275 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. B ,  D >. }  =  { D } )
74, 6uneq12d 3201 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )  =  ( { C }  u.  { D } ) )
8 rnun 4917 . . 3  |-  ran  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )  =  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )
9 df-pr 3504 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
107, 8, 93eqtr4g 2175 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } )  =  { C ,  D } )
112, 10syl5eq 2162 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 103    = wceq 1316    e. wcel 1465    u. cun 3039   {csn 3497   {cpr 3498   <.cop 3500   ran crn 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by: (None)
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