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Theorem nff 5329
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 5187 . 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 5279 . . 3  |-  F/ x  F  Fn  A
52nfrn 4844 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3131 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1552 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1461 1  |-  F/ x  F : A --> B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1447   F/_wnfc 2293    C_ wss 3112   ran crn 4600    Fn wfn 5178   -->wf 5179
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-fun 5185  df-fn 5186  df-f 5187
This theorem is referenced by:  nff1  5386
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