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Theorem nff1o 5765
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 5544 . 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 5730 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5745 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1988 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1690 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370   F/wnf 1661   F/_wnfc 2550   -1-1->wf1 5534   -onto->wfo 5535   -1-1-onto->wf1o 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ral 2713  df-rab 2717  df-v 3018  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-br 4360  df-opab 4419  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544
This theorem is referenced by:  nfiso  6167  nfsum1  13692  nfsum  13693  nfcprod1  13900  nfcprod  13901  esumiun  28860  wessf1ornlem  37357  stoweidlem35  37773
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