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Theorem 1stnpr 6785
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 6783 . 2  |-  ( 1st `  A )  =  U. dom  { A }
2 dmsnn0 5459 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
32biimpri 206 . . . . 5  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
43necon1bi 2674 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
54unieqd 4240 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
6 uni0 4257 . . 3  |-  U. (/)  =  (/)
75, 6syl6eq 2498 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  (/) )
81, 7syl5eq 2494 1  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 1st `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093   (/)c0 3767   {csn 4010   U.cuni 4230    X. cxp 4983   dom cdm 4985   ` cfv 5574   1stc1st 6779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fv 5582  df-1st 6781
This theorem is referenced by: (None)
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