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Theorem brtpos0 6478
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 6480. (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 6477 . 2  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  ( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) ) )
2 ssun2 3503 . . . . 5  |-  { (/) } 
C_  ( `' dom  F  u.  { (/) } )
3 0ex 4331 . . . . . 6  |-  (/)  e.  _V
43snid 3833 . . . . 5  |-  (/)  e.  { (/)
}
52, 4sselii 3337 . . . 4  |-  (/)  e.  ( `' dom  F  u.  { (/)
} )
65biantrur 493 . . 3  |-  ( U. `' { (/) } F A  <-> 
( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) )
7 cnvsn0 5330 . . . . . 6  |-  `' { (/)
}  =  (/)
87unieqi 4017 . . . . 5  |-  U. `' { (/) }  =  U. (/)
9 uni0 4034 . . . . 5  |-  U. (/)  =  (/)
108, 9eqtri 2455 . . . 4  |-  U. `' { (/) }  =  (/)
1110breq1i 4211 . . 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 1725    u. cun 3310   (/)c0 3620   {csn 3806   U.cuni 4007   class class class wbr 4204   `'ccnv 4869   dom cdm 4870  tpos ctpos 6470
This theorem is referenced by:  reldmtpos  6479  brtpos  6480  tpostpos  6491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-tpos 6471
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