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Theorem nffo 5777
Description: Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nffo.1  |-  F/_ x F
nffo.2  |-  F/_ x A
nffo.3  |-  F/_ x B
Assertion
Ref Expression
nffo  |-  F/ x  F : A -onto-> B

Proof of Theorem nffo
StepHypRef Expression
1 df-fo 5575 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
2 nffo.1 . . . 4  |-  F/_ x F
3 nffo.2 . . . 4  |-  F/_ x A
42, 3nffn 5658 . . 3  |-  F/ x  F  Fn  A
52nfrn 5066 . . . 4  |-  F/_ x ran  F
6 nffo.3 . . . 4  |-  F/_ x B
75, 6nfeq 2575 . . 3  |-  F/ x ran  F  =  B
84, 7nfan 1956 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  =  B )
91, 8nfxfr 1666 1  |-  F/ x  F : A -onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1405   F/wnf 1637   F/_wnfc 2550   ran crn 4824    Fn wfn 5564   -onto->wfo 5567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-fun 5571  df-fn 5572  df-fo 5575
This theorem is referenced by:  nff1o  5797
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