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Theorem nff1o 5365
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 5130 . 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 5326 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5344 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1544 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1450 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1436   F/_wnfc 2268   -1-1->wf1 5120   -onto->wfo 5121   -1-1-onto->wf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  nfiso  5707  nfsum1  11137  nfsum  11138  nfcprod1  11335  nfcprod  11336
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