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Theorem nfpconfp 32642
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 3961 . . 3 (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2 fnelfp 7195 . . . . 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 4237 . . . . . . . 8 (𝐹 ∩ I ) ⊆ 𝐹
5 dmss 5913 . . . . . . . 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 4026 . . . . . 6 (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
98sseld 3982 . . . . 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 7197 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑥) ≠ 𝑥))
1211notbid 318 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹𝑥) ≠ 𝑥))
13 nne 2944 . . . . . 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 2735 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 1540  wcel 2108  wne 2940  cdif 3948  cin 3950  wss 3951   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-11 2157  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-nfc 2892  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-mpt 5226  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:  symgcom2  33104
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