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Theorem nff1o 5635
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 5424 . 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 5600 . . 3  |-  F/ x  F : A -1-1-> B
62, 3, 4nffo 5615 . . 3  |-  F/ x  F : A -onto-> B
75, 6nfan 1842 . 2  |-  F/ x
( F : A -1-1-> B  /\  F : A -onto-> B )
81, 7nfxfr 1576 1  |-  F/ x  F : A -1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    /\ wa 359   F/wnf 1550   F/_wnfc 2531   -1-1->wf1 5414   -onto->wfo 5415   -1-1-onto->wf1o 5416
This theorem is referenced by:  nfiso  6007  nfsum1  12443  nfsum  12444  nfcprod1  25193  nfcprod  25194  stoweidlem35  27655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424
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