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Theorem nff 5954
 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 5808 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 5901 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 5289 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3561 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1816 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1771 1 𝑥 𝐹:𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383  Ⅎwnf 1699  Ⅎwnfc 2738   ⊆ wss 3540  ran crn 5039   Fn wfn 5799  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808 This theorem is referenced by:  nff1  6012  nfwrd  13188  lfgrnloop  25791  fcomptf  28840  aciunf1lem  28844  esumfzf  29458  esumfsup  29459  poimirlem24  32603  sdclem1  32709  dffo3f  38359  fmuldfeqlem1  38649  fnlimfvre  38741  dvnmul  38833  stoweidlem53  38946  stoweidlem54  38947  stoweidlem57  38950  sge0iunmpt  39311
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