Theorem nff1 5090

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 4907	. 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 5043	. . 3 |- F/ x F : A --> B
6	2	nfcnv 4514	. . . 4 |- F/_ x `' F
7	6	nffun 4924	. . 3 |- F/ x Fun `' F
8	5, 7	nfan 1457	. 2 |- F/ x ( F : A --> B /\ Fun `' F )
9	1, 8	nfxfr 1363	1 |- F/ x F : A -1-1-> B

Syntax hints:   /\ wa 97   F/wnf 1349   F/_wnfc 2165   `'ccnv 4344   Fun wfun 4896   -->wf 4898   -1-1->wf1 4899

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907

This theorem is referenced by:  nff1o  5124