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Theorem nff 5740
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 5603 . 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 5688 . . 3  |-  F/ x  F  Fn  A
52nfrn 5094 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3458 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1985 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1693 1  |-  F/ x  F : A --> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371   F/wnf 1664   F/_wnfc 2571    C_ wss 3437   ran crn 4852    Fn wfn 5594   -->wf 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-fun 5601  df-fn 5602  df-f 5603
This theorem is referenced by:  nff1  5792  nfwrd  12693  fcomptf  28256  aciunf1lem  28260  esumfzf  28892  esumfsup  28893  poimirlem24  31922  sdclem1  32030  dffo3f  37344  fmuldfeqlem1  37524  dvnmul  37682  stoweidlem53  37778  stoweidlem54  37779  stoweidlem57  37782  sge0iunmpt  38092
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