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Theorem nff1 5227
Description: Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1.1  |-  F/_ x F
nff1.2  |-  F/_ x A
nff1.3  |-  F/_ x B
Assertion
Ref Expression
nff1  |-  F/ x  F : A -1-1-> B

Proof of Theorem nff1
StepHypRef Expression
1 df-f1 5033 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
2 nff1.1 . . . 4  |-  F/_ x F
3 nff1.2 . . . 4  |-  F/_ x A
4 nff1.3 . . . 4  |-  F/_ x B
52, 3, 4nff 5171 . . 3  |-  F/ x  F : A --> B
62nfcnv 4628 . . . 4  |-  F/_ x `' F
76nffun 5051 . . 3  |-  F/ x Fun  `' F
85, 7nfan 1503 . 2  |-  F/ x
( F : A --> B  /\  Fun  `' F
)
91, 8nfxfr 1409 1  |-  F/ x  F : A -1-1-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 103   F/wnf 1395   F/_wnfc 2216   `'ccnv 4451   Fun wfun 5022   -->wf 5024   -1-1->wf1 5025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033
This theorem is referenced by:  nff1o  5264
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