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Theorem nff 5655
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  |-  F/_ x F
nff.2  |-  F/_ x A
nff.3  |-  F/_ x B
Assertion
Ref Expression
nff  |-  F/ x  F : A --> B

Proof of Theorem nff
StepHypRef Expression
1 df-f 5522 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 nff.1 . . . 4  |-  F/_ x F
3 nff.2 . . . 4  |-  F/_ x A
42, 3nffn 5607 . . 3  |-  F/ x  F  Fn  A
52nfrn 5182 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3449 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1863 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1616 1  |-  F/ x  F : A --> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   F/wnf 1590   F/_wnfc 2599    C_ wss 3428   ran crn 4941    Fn wfn 5513   -->wf 5514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-fun 5520  df-fn 5521  df-f 5522
This theorem is referenced by:  nff1  5704  nfwrd  12360  fcomptf  26116  esumfzf  26654  esumfsup  26655  sdclem1  28779  fmuldfeqlem1  29903  stoweidlem53  29988  stoweidlem54  29989  stoweidlem57  29992
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