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Theorem fnelfp 7122
Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnelfp ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑋) = 𝑋))

Proof of Theorem fnelfp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fninfp 7121 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥})
21eleq2d 2820 . 2 (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥}))
3 fveq2 6843 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
4 id 22 . . . 4 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4eqeq12d 2749 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) = 𝑥 ↔ (𝐹𝑋) = 𝑋))
65elrab3 3647 . 2 (𝑋𝐴 → (𝑋 ∈ {𝑥𝐴 ∣ (𝐹𝑥) = 𝑥} ↔ (𝐹𝑋) = 𝑋))
72, 6sylan9bb 511 1 ((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑋) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3406  cin 3910   I cid 5531  dom cdm 5634   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  nfpconfp  31592  ismrcd1  41064  ismrcd2  41065  istopclsd  41066
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