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Theorem nff 5718
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 5583 . 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 5668 . . 3  |-  F/ x  F  Fn  A
52nfrn 5236 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3490 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1870 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1620 1  |-  F/ x  F : A --> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   F/wnf 1594   F/_wnfc 2608    C_ wss 3469   ran crn 4993    Fn wfn 5574   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583
This theorem is referenced by:  nff1  5770  nfwrd  12522  fcomptf  27025  esumfzf  27565  esumfsup  27566  sdclem1  29690  fmuldfeqlem1  30951  stoweidlem53  31172  stoweidlem54  31173  stoweidlem57  31176
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