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Theorem nff 6501
 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 6349 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 6442 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5812 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3945 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1901 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1854 1 𝑥 𝐹:𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2962   ⊆ wss 3919  ran crn 5544   Fn wfn 6340  ⟶wf 6341 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-fun 6347  df-fn 6348  df-f 6349 This theorem is referenced by:  nff1  6565  nfwrd  13897  lfgrnloop  26927  fcomptf  30422  aciunf1lem  30426  fnpreimac  30435  esumfzf  31413  esumfsup  31414  poimirlem24  35053  sdclem1  35153  dffo3f  41757  fmuldfeqlem1  42177  fnlimfvre  42269  dvnmul  42538  stoweidlem53  42648  stoweidlem54  42649  stoweidlem57  42652  sge0iunmpt  43010
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