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Theorem 1stnpr 6777
Description: Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
1stnpr  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 1st `  A )  =  (/) )

Proof of Theorem 1stnpr
StepHypRef Expression
1 1stval 6775 . 2  |-  ( 1st `  A )  =  U. dom  { A }
2 dmsnn0 5456 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
32biimpri 206 . . . . 5  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
43necon1bi 2687 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
54unieqd 4245 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
6 uni0 4262 . . 3  |-  U. (/)  =  (/)
75, 6syl6eq 2511 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  (/) )
81, 7syl5eq 2507 1  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 1st `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106   (/)c0 3783   {csn 4016   U.cuni 4235    X. cxp 4986   dom cdm 4988   ` cfv 5570   1stc1st 6771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773
This theorem is referenced by: (None)
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