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Theorem brtpos0 6954
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 6956. (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 6953 . 2  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  ( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) ) )
2 ssun2 3663 . . . . 5  |-  { (/) } 
C_  ( `' dom  F  u.  { (/) } )
3 0ex 4572 . . . . . 6  |-  (/)  e.  _V
43snid 4050 . . . . 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 5469 . . . . . 6  |-  `' { (/)
}  =  (/)
87unieqi 4249 . . . . 5  |-  U. `' { (/) }  =  U. (/)
9 uni0 4267 . . . . 5  |-  U. (/)  =  (/)
108, 9eqtri 2491 . . . 4  |-  U. `' { (/) }  =  (/)
1110breq1i 4449 . . 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 1762    u. cun 3469   (/)c0 3780   {csn 4022   U.cuni 4240   class class class wbr 4442   `'ccnv 4993   dom cdm 4994  tpos ctpos 6946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-fv 5589  df-tpos 6947
This theorem is referenced by:  reldmtpos  6955  brtpos  6956  tpostpos  6967
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