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Theorem nff 6659
Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nff.1 𝑥𝐹
nff.2 𝑥𝐴
nff.3 𝑥𝐵
Assertion
Ref Expression
nff 𝑥 𝐹:𝐴𝐵

Proof of Theorem nff
StepHypRef Expression
1 df-f 6495 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6596 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5903 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3934 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1902 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1855 1 𝑥 𝐹:𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 396  wnf 1785  wnfc 2885  wss 3908  ran crn 5631   Fn wfn 6486  wf 6487
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-dm 5640  df-rn 5641  df-fun 6493  df-fn 6494  df-f 6495
This theorem is referenced by:  nff1  6731  nfwrd  14358  lfgrnloop  27874  fcomptf  31371  aciunf1lem  31375  fnpreimac  31384  esumfzf  32441  esumfsup  32442  poimirlem24  35997  sdclem1  36097  dffo3f  43155  fmuldfeqlem1  43576  fnlimfvre  43668  dvnmul  43937  stoweidlem53  44047  stoweidlem54  44048  stoweidlem57  44051  sge0iunmpt  44412
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