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Theorem nff1 5628
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 5450 . 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 5580 . . 3  |-  F/ x  F : A --> B
62nfcnv 5042 . . . 4  |-  F/_ x `' F
76nffun 5467 . . 3  |-  F/ x Fun  `' F
85, 7nfan 1846 . 2  |-  F/ x
( F : A --> B  /\  Fun  `' F
)
91, 8nfxfr 1579 1  |-  F/ x  F : A -1-1-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 359   F/wnf 1553   F/_wnfc 2558   `'ccnv 4868   Fun wfun 5439   -->wf 5441   -1-1->wf1 5442
This theorem is referenced by:  nff1o  5663  iundom2g  8404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450
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