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Theorem nffun 6070
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 6049 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
2 nffun.1 . . . 4 𝑥𝐹
32nfrel 5359 . . 3 𝑥Rel 𝐹
42nfcnv 5454 . . . . 5 𝑥𝐹
52, 4nfco 5441 . . . 4 𝑥(𝐹𝐹)
6 nfcv 2900 . . . 4 𝑥 I
75, 6nfss 3735 . . 3 𝑥(𝐹𝐹) ⊆ I
83, 7nfan 1975 . 2 𝑥(Rel 𝐹 ∧ (𝐹𝐹) ⊆ I )
91, 8nfxfr 1926 1 𝑥Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 383  wnf 1855  wnfc 2887  wss 3713   I cid 5171  ccnv 5263  ccom 5268  Rel wrel 5269  Fun wfun 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-rel 5271  df-cnv 5272  df-co 5273  df-fun 6049
This theorem is referenced by:  nffn  6146  nff1  6258  fliftfun  6723  funimass4f  29744  nfdfat  41714
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