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Theorem nff 5269
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 5127 . 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 5219 . . 3  |-  F/ x  F  Fn  A
52nfrn 4784 . . . 4  |-  F/_ x ran  F
6 nff.3 . . . 4  |-  F/_ x B
75, 6nfss 3090 . . 3  |-  F/ x ran  F  C_  B
84, 7nfan 1544 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  C_  B
)
91, 8nfxfr 1450 1  |-  F/ x  F : A --> B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1436   F/_wnfc 2268    C_ wss 3071   ran crn 4540    Fn wfn 5118   -->wf 5119
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  nff1  5326
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