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Theorem nffo 5622
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 5427 . 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 5510 . . 3  |-  F/ x  F  Fn  A
52nfrn 5085 . . . 4  |-  F/_ x ran  F
6 nffo.3 . . . 4  |-  F/_ x B
75, 6nfeq 2589 . . 3  |-  F/ x ran  F  =  B
84, 7nfan 1861 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  =  B )
91, 8nfxfr 1615 1  |-  F/ x  F : A -onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369   F/wnf 1589   F/_wnfc 2569   ran crn 4844    Fn wfn 5416   -onto->wfo 5419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ral 2723  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-br 4296  df-opab 4354  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-fun 5423  df-fn 5424  df-fo 5427
This theorem is referenced by:  nff1o  5642
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