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Theorem nff1 5640
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 5462 . 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 5592 . . 3  |-  F/ x  F : A --> B
62nfcnv 5054 . . . 4  |-  F/_ x `' F
76nffun 5479 . . 3  |-  F/ x Fun  `' F
85, 7nfan 1847 . 2  |-  F/ x
( F : A --> B  /\  Fun  `' F
)
91, 8nfxfr 1580 1  |-  F/ x  F : A -1-1-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 360   F/wnf 1554   F/_wnfc 2561   `'ccnv 4880   Fun wfun 5451   -->wf 5453   -1-1->wf1 5454
This theorem is referenced by:  nff1o  5675  iundom2g  8420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462
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