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Theorem fnelnfp 7167
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 7166 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
21eleq2d 2811 . 2 (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥}))
3 fveq2 6881 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4neeq12d 2994 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑋) ≠ 𝑋))
65elrab3 3676 . 2 (𝑋𝐴 → (𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} ↔ (𝐹𝑋) ≠ 𝑋))
72, 6sylan9bb 509 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2932  {crab 3424  cdif 3937   I cid 5563  dom cdm 5666   Fn wfn 6528  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541
This theorem is referenced by:  f1omvdmvd  19348  f1omvdconj  19351  f1otrspeq  19352  pmtrfinv  19366  symggen  19375  psgnunilem1  19398  mdetdiaglem  22410  mdetralt  22420  mdetunilem7  22430  nfpconfp  32280  pmtrcnel  32677  pmtrcnel2  32678  pmtrcnelor  32679  cycpmrn  32729
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