MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nffun Structured version   Visualization version   GIF version

Theorem nffun 6457
Description: Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
Hypothesis
Ref Expression
nffun.1 𝑥𝐹
Assertion
Ref Expression
nffun 𝑥Fun 𝐹

Proof of Theorem nffun
StepHypRef Expression
1 df-fun 6435 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5690 . . 3 𝑥Rel 𝐹
42nfcnv 5787 . . . . 5 𝑥𝐹
52, 4nfco 5774 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2907 . . . 4 𝑥 I
75, 6nfss 3913 . . 3 𝑥(𝐹𝐹) ⊆ I
83, 7nfan 1902 . 2 𝑥(Rel 𝐹 ∧ (𝐹𝐹) ⊆ I )
91, 8nfxfr 1855 1 𝑥Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 396  wnf 1786  wnfc 2887  wss 3887   I cid 5488  ccnv 5588  ccom 5593  Rel wrel 5594  Fun wfun 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-fun 6435
This theorem is referenced by:  nffn  6532  nff1  6668  fliftfun  7183  funimass4f  30972  nfdfat  44619
  Copyright terms: Public domain W3C validator