MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfunsn Unicode version

Theorem nfunsn 5902
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 2313 . . . . . . 7
2 vex 3112 . . . . . . . . . 10
32brres 5285 . . . . . . . . 9
4 elsn 4043 . . . . . . . . . . 11
5 breq1 4455 . . . . . . . . . . 11
64, 5sylbi 195 . . . . . . . . . 10
76biimpac 486 . . . . . . . . 9
83, 7sylbi 195 . . . . . . . 8
98moimi 2340 . . . . . . 7
101, 9syl 16 . . . . . 6
11 tz6.12-2 5862 . . . . . 6
1210, 11nsyl4 142 . . . . 5
1312alrimiv 1719 . . . 4
14 relres 5306 . . . 4
1513, 14jctil 537 . . 3
16 dffun6 5608 . . 3
1715, 16sylibr 212 . 2
1817con1i 129 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  E!weu 2282  E*wmo 2283   c0 3784  {csn 4029   class class class wbr 4452  |`cres 5006  Relwrel 5009  Funwfun 5587  `cfv 5593
This theorem is referenced by:  fvfundmfvn0  5903  dffv2  5946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-res 5016  df-iota 5556  df-fun 5595  df-fv 5601
  Copyright terms: Public domain W3C validator