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Theorem nff 5313
 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 5171 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 nff.1 . . . 4 𝑥𝐹
3 nff.2 . . . 4 𝑥𝐴
42, 3nffn 5263 . . 3 𝑥 𝐹 Fn 𝐴
52nfrn 4828 . . . 4 𝑥ran 𝐹
6 nff.3 . . . 4 𝑥𝐵
75, 6nfss 3121 . . 3 𝑥ran 𝐹𝐵
84, 7nfan 1545 . 2 𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹𝐵)
91, 8nfxfr 1454 1 𝑥 𝐹:𝐴𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 103  Ⅎwnf 1440  Ⅎwnfc 2286   ⊆ wss 3102  ran crn 4584   Fn wfn 5162  ⟶wf 5163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-fun 5169  df-fn 5170  df-f 5171 This theorem is referenced by:  nff1  5370
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