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Theorem nffo 5785
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 5585 . 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 5668 . . 3  |-  F/ x  F  Fn  A
52nfrn 5236 . . . 4  |-  F/_ x ran  F
6 nffo.3 . . . 4  |-  F/_ x B
75, 6nfeq 2633 . . 3  |-  F/ x ran  F  =  B
84, 7nfan 1870 . 2  |-  F/ x
( F  Fn  A  /\  ran  F  =  B )
91, 8nfxfr 1620 1  |-  F/ x  F : A -onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374   F/wnf 1594   F/_wnfc 2608   ran crn 4993    Fn wfn 5574   -onto->wfo 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-fo 5585
This theorem is referenced by:  nff1o  5805
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