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Theorem nffun 6520
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 6494 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5732 . . 3 𝑥Rel 𝐹
42nfcnv 5831 . . . . 5 𝑥𝐹
52, 4nfco 5818 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2906 . . . 4 𝑥 I
75, 6nfss 3935 . . 3 𝑥(𝐹𝐹) ⊆ I
83, 7nfan 1903 . 2 𝑥(Rel 𝐹 ∧ (𝐹𝐹) ⊆ I )
91, 8nfxfr 1856 1 𝑥Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 397  wnf 1786  wnfc 2886  wss 3909   I cid 5528  ccnv 5630  ccom 5635  Rel wrel 5636  Fun wfun 6486
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6494
This theorem is referenced by:  nffn  6597  nff1  6732  fliftfun  7252  funimass4f  31336  nfdfat  45077
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