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Theorem fnelnfp 7122
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 7121 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
21eleq2d 2823 . 2 (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥}))
3 fveq2 6842 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4neeq12d 3005 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑋) ≠ 𝑋))
65elrab3 3646 . 2 (𝑋𝐴 → (𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} ↔ (𝐹𝑋) ≠ 𝑋))
72, 6sylan9bb 510 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943  {crab 3407  cdif 3907   I cid 5530  dom cdm 5633   Fn wfn 6491  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504
This theorem is referenced by:  f1omvdmvd  19223  f1omvdconj  19226  f1otrspeq  19227  pmtrfinv  19241  symggen  19250  psgnunilem1  19273  mdetdiaglem  21945  mdetralt  21955  mdetunilem7  21965  nfpconfp  31493  pmtrcnel  31884  pmtrcnel2  31885  pmtrcnelor  31886  cycpmrn  31936
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