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Theorem nff1o 5124
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 4909 . 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 5090 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5105 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1457 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1363 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 97   F/wnf 1349   F/_wnfc 2165   -1-1->wf1 4899   -onto->wfo 4900   -1-1-onto->wf1o 4901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909
This theorem is referenced by:  nfiso  5446  nfsum1  9849  nfsum  9850
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