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

Theorem nff1 6551
 Description: Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1.1 𝑥𝐹
nff1.2 𝑥𝐴
nff1.3 𝑥𝐵
Assertion
Ref Expression
nff1 𝑥 𝐹:𝐴1-1𝐵

Proof of Theorem nff1
StepHypRef Expression
1 df-f1 6333 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
2 nff1.1 . . . 4 𝑥𝐹
3 nff1.2 . . . 4 𝑥𝐴
4 nff1.3 . . . 4 𝑥𝐵
52, 3, 4nff 6487 . . 3 𝑥 𝐹:𝐴𝐵
62nfcnv 5717 . . . 4 𝑥𝐹
76nffun 6351 . . 3 𝑥Fun 𝐹
85, 7nfan 1900 . 2 𝑥(𝐹:𝐴𝐵 ∧ Fun 𝐹)
91, 8nfxfr 1854 1 𝑥 𝐹:𝐴1-1𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2939  ◡ccnv 5522  Fun wfun 6322  ⟶wf 6324  –1-1→wf1 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333 This theorem is referenced by:  nff1o  6592  iundom2g  9955
 Copyright terms: Public domain W3C validator