Theorem nff1 5431

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 5233	. 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 5374	. . 3 |- F/ x F : A --> B
6	2	nfcnv 4818	. . . 4 |- F/_ x `' F
7	6	nffun 5251	. . 3 |- F/ x Fun `' F
8	5, 7	nfan 1575	. 2 |- F/ x ( F : A --> B /\ Fun `' F )
9	1, 8	nfxfr 1484	1 |- F/ x F : A -1-1-> B

Syntax hints:   /\ wa 104   F/wnf 1470   F/_wnfc 2316   `'ccnv 4637   Fun wfun 5222   -->wf 5224   -1-1->wf1 5225

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233

This theorem is referenced by:  nff1o  5471