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

Theorem nffun 6377
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 6356 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5653 . . 3 𝑥Rel 𝐹
42nfcnv 5748 . . . . 5 𝑥𝐹
52, 4nfco 5735 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2982 . . . 4 𝑥 I
75, 6nfss 3964 . . 3 𝑥(𝐹𝐹) ⊆ I
83, 7nfan 1893 . 2 𝑥(Rel 𝐹 ∧ (𝐹𝐹) ⊆ I )
91, 8nfxfr 1846 1 𝑥Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 396  wnf 1777  wnfc 2966  wss 3940   I cid 5458  ccnv 5553  ccom 5558  Rel wrel 5559  Fun wfun 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-fun 6356
This theorem is referenced by:  nffn  6451  nff1  6572  fliftfun  7059  funimass4f  30316  nfdfat  43211
  Copyright terms: Public domain W3C validator