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Theorem nff1o 5829
Description: Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1o.1  |-  F/_ x F
nff1o.2  |-  F/_ x A
nff1o.3  |-  F/_ x B
Assertion
Ref Expression
nff1o  |-  F/ x  F : A -1-1-onto-> B

Proof of Theorem nff1o
StepHypRef Expression
1 df-f1o 5608 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 nff1o.1 . . . 4  |-  F/_ x F
3 nff1o.2 . . . 4  |-  F/_ x A
4 nff1o.3 . . . 4  |-  F/_ x B
52, 3, 4nff1 5794 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5809 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1986 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1692 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370   F/wnf 1663   F/_wnfc 2577   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600
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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
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 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608
This theorem is referenced by:  nfiso  6230  nfsum1  13734  nfsum  13735  nfcprod1  13942  nfcprod  13943  esumiun  28754  wessf1ornlem  37085  stoweidlem35  37468
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