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Theorem brtpos0 6852
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 6854. (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 6851 . 2  |-  ( A  e.  V  ->  ( (/)tpos  F A  <->  ( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) ) )
2 ssun2 3618 . . . . 5  |-  { (/) } 
C_  ( `' dom  F  u.  { (/) } )
3 0ex 4520 . . . . . 6  |-  (/)  e.  _V
43snid 4003 . . . . 5  |-  (/)  e.  { (/)
}
52, 4sselii 3451 . . . 4  |-  (/)  e.  ( `' dom  F  u.  { (/)
} )
65biantrur 506 . . 3  |-  ( U. `' { (/) } F A  <-> 
( (/)  e.  ( `' dom  F  u.  { (/)
} )  /\  U. `' { (/) } F A ) )
7 cnvsn0 5405 . . . . . 6  |-  `' { (/)
}  =  (/)
87unieqi 4198 . . . . 5  |-  U. `' { (/) }  =  U. (/)
9 uni0 4216 . . . . 5  |-  U. (/)  =  (/)
108, 9eqtri 2480 . . . 4  |-  U. `' { (/) }  =  (/)
1110breq1i 4397 . . 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 1758    u. cun 3424   (/)c0 3735   {csn 3975   U.cuni 4189   class class class wbr 4390   `'ccnv 4937   dom cdm 4938  tpos ctpos 6844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-fv 5524  df-tpos 6845
This theorem is referenced by:  reldmtpos  6853  brtpos  6854  tpostpos  6865
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