Theorem nff1 5501

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 5295	. 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 5442	. . 3 |- F/ x F : A --> B
6	2	nfcnv 4875	. . . 4 |- F/_ x `' F
7	6	nffun 5313	. . 3 |- F/ x Fun `' F
8	5, 7	nfan 1589	. 2 |- F/ x ( F : A --> B /\ Fun `' F )
9	1, 8	nfxfr 1498	1 |- F/ x F : A -1-1-> B

Syntax hints:   /\ wa 104   F/wnf 1484   F/_wnfc 2337   `'ccnv 4692   Fun wfun 5284   -->wf 5286   -1-1->wf1 5287

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295

This theorem is referenced by:  nff1o  5542