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Theorem nffun 6519
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 6493 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5731 . . 3 𝑥Rel 𝐹
42nfcnv 5830 . . . . 5 𝑥𝐹
52, 4nfco 5817 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2905 . . . 4 𝑥 I
75, 6nfss 3934 . . 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 1785  wnfc 2885  wss 3908   I cid 5527  ccnv 5629  ccom 5634  Rel wrel 5635  Fun wfun 6485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-rel 5637  df-cnv 5638  df-co 5639  df-fun 6493
This theorem is referenced by:  nffn  6596  nff1  6731  fliftfun  7251  funimass4f  31348  nfdfat  45108
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