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

Theorem fnelnfp 6326
Description: Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
fnelnfp ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))

Proof of Theorem fnelnfp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fndifnfp 6325 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
21eleq2d 2673 . 2 (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥}))
3 fveq2 6088 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4neeq12d 2843 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑋) ≠ 𝑋))
65elrab3 3332 . 2 (𝑋𝐴 → (𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} ↔ (𝐹𝑋) ≠ 𝑋))
72, 6sylan9bb 732 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  {crab 2900  cdif 3537   I cid 4938  dom cdm 5028   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4704  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-opab 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798
This theorem is referenced by:  f1omvdmvd  17635  f1omvdconj  17638  f1otrspeq  17639  pmtrfinv  17653  symggen  17662  psgnunilem1  17685  mdetdiaglem  20171  mdetralt  20181  mdetunilem7  20191
  Copyright terms: Public domain W3C validator