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Theorem fvfundmfvn0 5903
Description: If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
fvfundmfvn0

Proof of Theorem fvfundmfvn0
StepHypRef Expression
1 ianor 488 . . 3
2 ndmfv 5895 . . . 4
3 nfunsn 5902 . . . 4
42, 3jaoi 379 . . 3
51, 4sylbi 195 . 2
65necon1ai 2688 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652   c0 3784  {csn 4029  domcdm 5004  |`cres 5006  Funwfun 5587  `cfv 5593
This theorem is referenced by:  fvn0ssdmfun  6022  fvn0fvelrn  6088  usgranloopv  24378  afvpcfv0  32231  afvfvn0fveq  32235  afv0nbfvbi  32236  ovn0dmfun  32452
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-8 1820  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-pow 4630  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-dm 5014  df-res 5016  df-iota 5556  df-fun 5595  df-fv 5601
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