Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvsnn Unicode version

Theorem fvsnn 25217
Description: Value when  C doesn't belong to the domain. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
fvsnn  |-  ( C  =/=  A  ->  ( { <. A ,  B >. } `  C )  =  (/) )

Proof of Theorem fvsnn
StepHypRef Expression
1 dmsnopss 5161 . . . . 5  |-  dom  { <. A ,  B >. } 
C_  { A }
21sseli 3189 . . . 4  |-  ( C  e.  dom  { <. A ,  B >. }  ->  C  e.  { A }
)
3 elsni 3677 . . . 4  |-  ( C  e.  { A }  ->  C  =  A )
42, 3syl 15 . . 3  |-  ( C  e.  dom  { <. A ,  B >. }  ->  C  =  A )
54necon3ai 2499 . 2  |-  ( C  =/=  A  ->  -.  C  e.  dom  { <. A ,  B >. } )
6 ndmfv 5568 . 2  |-  ( -.  C  e.  dom  { <. A ,  B >. }  ->  ( { <. A ,  B >. } `  C )  =  (/) )
75, 6syl 15 1  |-  ( C  =/=  A  ->  ( { <. A ,  B >. } `  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {csn 3653   <.cop 3656   dom cdm 4705   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-dm 4715  df-iota 5235  df-fv 5279
  Copyright terms: Public domain W3C validator