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Theorem nffun 6572
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 6546 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5780 . . 3 𝑥Rel 𝐹
42nfcnv 5879 . . . . 5 𝑥𝐹
52, 4nfco 5866 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2904 . . . 4 𝑥 I
75, 6nfss 3975 . . 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 2884  wss 3949   I cid 5574  ccnv 5676  ccom 5681  Rel wrel 5682  Fun wfun 6538
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546
This theorem is referenced by:  nffn  6649  nff1  6786  fliftfun  7309  funimass4f  31861  nfdfat  45835
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