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Theorem brtpos0 6980
Description: The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows us to eliminate sethood hypotheses on  A ,  B in brtpos 6982. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 6979 . 2  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  ( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) ) )
2 ssun2 3664 . . . . 5  |-  { (/) } 
C_  ( `' dom  F  u.  { (/) } )
3 0ex 4587 . . . . . 6  |-  (/)  e.  _V
43snid 4060 . . . . 5  |-  (/)  e.  { (/)
}
52, 4sselii 3496 . . . 4  |-  (/)  e.  ( `' dom  F  u.  { (/)
} )
65biantrur 506 . . 3  |-  ( U. `' { (/) } F A  <-> 
( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) )
7 cnvsn0 5482 . . . . . 6  |-  `' { (/)
}  =  (/)
87unieqi 4260 . . . . 5  |-  U. `' { (/) }  =  U. (/)
9 uni0 4278 . . . . 5  |-  U. (/)  =  (/)
108, 9eqtri 2486 . . . 4  |-  U. `' { (/) }  =  (/)
1110breq1i 4463 . . 3  |-  ( U. `' { (/) } F A  <->  (/) F A )
126, 11bitr3i 251 . 2  |-  ( (
(/)  e.  ( `' dom  F  u.  { (/) } )  /\  U. `' { (/) } F A )  <->  (/) F A )
131, 12syl6bb 261 1  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1819    u. cun 3469   (/)c0 3793   {csn 4032   U.cuni 4251   class class class wbr 4456   `'ccnv 5007   dom cdm 5008  tpos ctpos 6972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-tpos 6973
This theorem is referenced by:  reldmtpos  6981  brtpos  6982  tpostpos  6993
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