Theorem nff1 5391

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

Step	Hyp	Ref	Expression
1		df-f1 5193	. 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
5	2, 3, 4	nff 5334	. . 3 |- F/ x F : A --> B
6	2	nfcnv 4783	. . . 4 |- F/_ x `' F
7	6	nffun 5211	. . 3 |- F/ x Fun `' F
8	5, 7	nfan 1553	. 2 |- F/ x ( F : A --> B /\ Fun `' F )
9	1, 8	nfxfr 1462	1 |- F/ x F : A -1-1-> B

Syntax hints:   /\ wa 103   F/wnf 1448   F/_wnfc 2295   `'ccnv 4603   Fun wfun 5182   -->wf 5184   -1-1->wf1 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193

This theorem is referenced by:  nff1o  5430