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Theorem nff1o 5471
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 5235 . 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 5431 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5449 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1575 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1484 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 104   F/wnf 1470   F/_wnfc 2316   -1-1->wf1 5225   -onto->wfo 5226   -1-1-onto->wf1o 5227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235
This theorem is referenced by:  nfiso  5820  nfsum1  11378  nfsum  11379  nfcprod1  11576  nfcprod  11577
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