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Theorem nfpconfp 32648
Description: The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.)
Assertion
Ref Expression
nfpconfp (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

Proof of Theorem nfpconfp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eldif 3972 . . 3 (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2 fnelfp 7194 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
32pm5.32da 579 . . . 4 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑥)))
4 inss1 4244 . . . . . . . 8 (𝐹 ∩ I ) ⊆ 𝐹
5 dmss 5915 . . . . . . . 8 ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
64, 5ax-mp 5 . . . . . . 7 dom (𝐹 ∩ I ) ⊆ dom 𝐹
7 fndm 6671 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
86, 7sseqtrid 4047 . . . . . 6 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
98sseld 3993 . . . . 5 (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) → 𝑥𝐴))
109pm4.71rd 562 . . . 4 (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝑥𝐴𝑥 ∈ dom (𝐹 ∩ I ))))
11 fnelnfp 7196 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
1211notbid 318 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹𝑥) ≠ 𝑥))
13 nne 2941 . . . . . 6 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
1412, 13bitrdi 287 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) = 𝑥))
1514pm5.32da 579 . . . 4 (𝐹 Fn 𝐴 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑥)))
163, 10, 153bitr4rd 312 . . 3 (𝐹 Fn 𝐴 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
171, 16bitrid 283 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
1817eqrdv 2732 1 (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  cdif 3959  cin 3961  wss 3962   I cid 5581  dom cdm 5688   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570
This theorem is referenced by:  symgcom2  33086
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