MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nff1 Structured version   Unicode version

Theorem nff1 5769
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
StepHypRef Expression
1 df-f1 5583 . 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
52, 3, 4nff 5717 . . 3  |-  F/ x  F : A --> B
62nfcnv 5171 . . . 4  |-  F/_ x `' F
76nffun 5600 . . 3  |-  F/ x Fun  `' F
85, 7nfan 1914 . 2  |-  F/ x
( F : A --> B  /\  Fun  `' F
)
91, 8nfxfr 1632 1  |-  F/ x  F : A -1-1-> B
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   F/wnf 1603   F/_wnfc 2591   `'ccnv 4988   Fun wfun 5572   -->wf 5574   -1-1->wf1 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583
This theorem is referenced by:  nff1o  5804  iundom2g  8918
  Copyright terms: Public domain W3C validator