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Theorem rnpropg 4800
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 3382 . . 3  |-  { <. A ,  C >. ,  <. B ,  D >. }  =  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )
21rneqi 4562 . 2  |-  ran  { <. A ,  C >. , 
<. B ,  D >. }  =  ran  ( {
<. A ,  C >. }  u.  { <. B ,  D >. } )
3 rnsnopg 4799 . . . . 5  |-  ( A  e.  V  ->  ran  {
<. A ,  C >. }  =  { C }
)
43adantr 261 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. A ,  C >. }  =  { C } )
5 rnsnopg 4799 . . . . 5  |-  ( B  e.  W  ->  ran  {
<. B ,  D >. }  =  { D }
)
65adantl 262 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  { <. B ,  D >. }  =  { D } )
74, 6uneq12d 3098 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )  =  ( { C }  u.  { D } ) )
8 rnun 4732 . . 3  |-  ran  ( { <. A ,  C >. }  u.  { <. B ,  D >. } )  =  ( ran  { <. A ,  C >. }  u.  ran  { <. B ,  D >. } )
9 df-pr 3382 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
107, 8, 93eqtr4g 2097 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ran  ( { <. A ,  C >. }  u.  {
<. B ,  D >. } )  =  { C ,  D } )
112, 10syl5eq 2084 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 97    = wceq 1243    e. wcel 1393    u. cun 2915   {csn 3375   {cpr 3376   <.cop 3378   ran crn 4346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356
This theorem is referenced by: (None)
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