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Theorem fnelnfp 7197
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 7196 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
21eleq2d 2827 . 2 (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥}))
3 fveq2 6906 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4neeq12d 3002 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑋) ≠ 𝑋))
65elrab3 3693 . 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 206  wa 395   = wceq 1540  wcel 2108  wne 2940  {crab 3436  cdif 3948   I cid 5577  dom cdm 5685   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  f1omvdmvd  19461  f1omvdconj  19464  f1otrspeq  19465  pmtrfinv  19479  symggen  19488  psgnunilem1  19511  mdetdiaglem  22604  mdetralt  22614  mdetunilem7  22624  nfpconfp  32642  pmtrcnel  33109  pmtrcnel2  33110  pmtrcnelor  33111  cycpmrn  33163
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