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Theorem nfunsn 5790
 Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
nfunsn

Proof of Theorem nfunsn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eumo 2327 . . . . . . 7
2 vex 2965 . . . . . . . . . 10
32brres 5181 . . . . . . . . 9
4 elsn 3853 . . . . . . . . . . 11
5 breq1 4240 . . . . . . . . . . 11
64, 5sylbi 189 . . . . . . . . . 10
76biimpac 474 . . . . . . . . 9
83, 7sylbi 189 . . . . . . . 8
98moimi 2334 . . . . . . 7
101, 9syl 16 . . . . . 6
11 tz6.12-2 5748 . . . . . 6
1210, 11nsyl4 137 . . . . 5
1312alrimiv 1642 . . . 4
14 relres 5203 . . . 4
1513, 14jctil 525 . . 3
16 dffun6 5498 . . 3
1715, 16sylibr 205 . 2
1817con1i 124 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550   wceq 1653   wcel 1727  weu 2287  wmo 2288  c0 3613  csn 3838   class class class wbr 4237   cres 4909   wrel 4912   wfun 5477  cfv 5483 This theorem is referenced by:  fvfundmfvn0  5791  dffv2  5825 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-res 4919  df-iota 5447  df-fun 5485  df-fv 5491
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