Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nff1o Structured version   Visualization version   GIF version

Theorem nff1o 6595
 Description: Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
Hypotheses
Ref Expression
nff1o.1 𝑥𝐹
nff1o.2 𝑥𝐴
nff1o.3 𝑥𝐵
Assertion
Ref Expression
nff1o 𝑥 𝐹:𝐴1-1-onto𝐵

Proof of Theorem nff1o
StepHypRef Expression
1 df-f1o 6341 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 nff1o.1 . . . 4 𝑥𝐹
3 nff1o.2 . . . 4 𝑥𝐴
4 nff1o.3 . . . 4 𝑥𝐵
52, 3, 4nff1 6554 . . 3 𝑥 𝐹:𝐴1-1𝐵
62, 3, 4nffo 6571 . . 3 𝑥 𝐹:𝐴onto𝐵
75, 6nfan 1900 . 2 𝑥(𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵)
81, 7nfxfr 1854 1 𝑥 𝐹:𝐴1-1-onto𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2960  –1-1→wf1 6331  –onto→wfo 6332  –1-1-onto→wf1o 6333 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 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 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341 This theorem is referenced by:  nfiso  7059  nfsum1  15037  nfsum  15038  nfsumOLD  15039  nfcprod1  15255  nfcprod  15256  fsumiunle  30555  esumiun  31427  stoweidlem35  42620
 Copyright terms: Public domain W3C validator