Theorem nff1o 5632

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

Step	Hyp	Ref	Expression
1		df-f1o 5379	. 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
5	2, 3, 4	nff1 5591	. . 3 |- F/ x F : A -1-1-> B
6	2, 3, 4	nffo 5609	. . 3 |- F/ x F : A -onto-> B
7	5, 6	nfan 1618	. 2 |- F/ x ( F : A -1-1-> B /\ F : A -onto-> B )
8	1, 7	nfxfr 1527	1 |- F/ x F : A -1-1-onto-> B

Syntax hints:   /\ wa 104   F/wnf 1513   F/_wnfc 2379   -1-1->wf1 5369   -onto->wfo 5370   -1-1-onto->wf1o 5371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379

This theorem is referenced by:  nfiso  6002  nfsum1  12100  nfsum  12101  nfcprod1  12299  nfcprod  12300