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Theorem rnpropg 4910
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 3453 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
21rneqi 4663 . 2  |-  ran  { <. A ,  C >. , 
<. B ,  D >. }  =  ran  ( {
<. A ,  C >. }  u.  { <. B ,  D >. } )
3 rnsnopg 4909 . . . . 5  |-  ( A  e.  V  ->  ran  {
<. A ,  C >. }  =  { C }
)
43adantr 270 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. }  =  { C } )
5 rnsnopg 4909 . . . . 5  |-  ( B  e.  W  ->  ran  {
<. B ,  D >. }  =  { D }
)
65adantl 271 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. B ,  D >. }  =  { D } )
74, 6uneq12d 3155 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )  =  ( { C }  u.  { D } ) )
8 rnun 4840 . . 3  |-  ran  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )  =  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )
9 df-pr 3453 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
107, 8, 93eqtr4g 2145 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } )  =  { C ,  D } )
112, 10syl5eq 2132 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 1289    e. wcel 1438    u. cun 2997   {csn 3446   {cpr 3447   <.cop 3449   ran crn 4439
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by: (None)
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