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Theorem nffun 6356
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 6335 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5619 . . 3 𝑥Rel 𝐹
42nfcnv 5715 . . . . 5 𝑥𝐹
52, 4nfco 5702 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2899 . . . 4 𝑥 I
75, 6nfss 3867 . . 3 𝑥(𝐹𝐹) ⊆ I
83, 7nfan 1905 . 2 𝑥(Rel 𝐹 ∧ (𝐹𝐹) ⊆ I )
91, 8nfxfr 1859 1 𝑥Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 399  wnf 1790  wnfc 2879  wss 3841   I cid 5424  ccnv 5518  ccom 5523  Rel wrel 5524  Fun wfun 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-v 3399  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-rel 5526  df-cnv 5527  df-co 5528  df-fun 6335
This theorem is referenced by:  nffn  6431  nff1  6566  fliftfun  7072  funimass4f  30538  nfdfat  44136
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