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Theorem 1stnpr 24093
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 6351 . 2  |-  ( 1st `  A )  =  U. dom  { A }
2 dmsnn0 5335 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  <->  dom  { A }  =/=  (/) )
32biimpri 198 . . . . 5  |-  ( dom 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
43necon1bi 2647 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  dom  { A }  =  (/) )
54unieqd 4026 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  U. (/) )
6 uni0 4042 . . 3  |-  U. (/)  =  (/)
75, 6syl6eq 2484 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. dom  { A }  =  (/) )
81, 7syl5eq 2480 1  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 1st `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   (/)c0 3628   {csn 3814   U.cuni 4015    X. cxp 4876   dom cdm 4878   ` cfv 5454   1stc1st 6347
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-1st 6349
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