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Theorem fvsnn 25114
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 5145 . . . . 5  |-  dom  { <. A ,  B >. } 
C_  { A }
21sseli 3176 . . . 4  |-  ( C  e.  dom  { <. A ,  B >. }  ->  C  e.  { A }
)
3 elsni 3664 . . . 4  |-  ( C  e.  { A }  ->  C  =  A )
42, 3syl 15 . . 3  |-  ( C  e.  dom  { <. A ,  B >. }  ->  C  =  A )
54necon3ai 2486 . 2  |-  ( C  =/=  A  ->  -.  C  e.  dom  { <. A ,  B >. } )
6 ndmfv 5552 . 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 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   {csn 3640   <.cop 3643   dom cdm 4689   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-iota 5219  df-fv 5263
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