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Theorem nffo 5806
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 5604 . 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 5687 . . 3  |-  F/ x  F  Fn  A
52nfrn 5093 . . . 4  |-  F/_ x ran  F
6 nffo.3 . . . 4  |-  F/_ x B
75, 6nfeq 2595 . . 3  |-  F/ x ran  F  =  B
84, 7nfan 1984 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  =  B )
91, 8nfxfr 1692 1  |-  F/ x  F : A -onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   F/wnf 1663   F/_wnfc 2570   ran crn 4851    Fn wfn 5593   -onto->wfo 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-fun 5600  df-fn 5601  df-fo 5604
This theorem is referenced by:  nff1o  5826  fompt  37317
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