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Theorem brtpos0 6424
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 6426. (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 6423 . 2  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  ( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) ) )
2 ssun2 3456 . . . . 5  |-  { (/) } 
C_  ( `' dom  F  u.  { (/) } )
3 0ex 4282 . . . . . 6  |-  (/)  e.  _V
43snid 3786 . . . . 5  |-  (/)  e.  { (/)
}
52, 4sselii 3290 . . . 4  |-  (/)  e.  ( `' dom  F  u.  { (/)
} )
65biantrur 493 . . 3  |-  ( U. `' { (/) } F A  <-> 
( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) )
7 cnvsn0 5280 . . . . . 6  |-  `' { (/)
}  =  (/)
87unieqi 3969 . . . . 5  |-  U. `' { (/) }  =  U. (/)
9 uni0 3986 . . . . 5  |-  U. (/)  =  (/)
108, 9eqtri 2409 . . . 4  |-  U. `' { (/) }  =  (/)
1110breq1i 4162 . . 3  |-  ( U. `' { (/) } F A  <->  (/) F A )
126, 11bitr3i 243 . 2  |-  ( (
(/)  e.  ( `' dom  F  u.  { (/) } )  /\  U. `' { (/) } F A )  <->  (/) F A )
131, 12syl6bb 253 1  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  (/) F A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    u. cun 3263   (/)c0 3573   {csn 3759   U.cuni 3959   class class class wbr 4155   `'ccnv 4819   dom cdm 4820  tpos ctpos 6416
This theorem is referenced by:  reldmtpos  6425  brtpos  6426  tpostpos  6437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-tpos 6417
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