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Theorem nfpconfp 30489
 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 3868 . . 3 (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2 fnelfp 6928 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
32pm5.32da 582 . . . 4 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑥)))
4 inss1 4133 . . . . . . . 8 (𝐹 ∩ I ) ⊆ 𝐹
5 dmss 5742 . . . . . . . 8 ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
64, 5ax-mp 5 . . . . . . 7 dom (𝐹 ∩ I ) ⊆ dom 𝐹
7 fndm 6436 . . . . . . 7 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
86, 7sseqtrid 3944 . . . . . 6 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
98sseld 3891 . . . . 5 (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) → 𝑥𝐴))
109pm4.71rd 566 . . . 4 (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝑥𝐴𝑥 ∈ dom (𝐹 ∩ I ))))
11 fnelnfp 6930 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
1211notbid 321 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹𝑥) ≠ 𝑥))
13 nne 2955 . . . . . 6 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
1412, 13bitrdi 290 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) = 𝑥))
1514pm5.32da 582 . . . 4 (𝐹 Fn 𝐴 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ (𝑥𝐴 ∧ (𝐹𝑥) = 𝑥)))
163, 10, 153bitr4rd 315 . . 3 (𝐹 Fn 𝐴 → ((𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
171, 16syl5bb 286 . 2 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
1817eqrdv 2756 1 (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∖ cdif 3855   ∩ cin 3857   ⊆ wss 3858   I cid 5429  dom cdm 5524   Fn wfn 6330  ‘cfv 6335 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343 This theorem is referenced by:  symgcom2  30879
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