Theorem nfpconfp 33048

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.

Step	Hyp	Ref	Expression
1		eldif 3916	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2		fnelfp 7179	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
3	2	pm5.32da 590	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
4		inss1 4189	. . . . . . . 8 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
5		dmss 5894	. . . . . . . 8 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
7		fndm 6642	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	6, 7	sseqtrid 3980	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
9	8	sseld 3937	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) → 𝑥 ∈ 𝐴))
10	9	pm4.71rd 572	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I ))))
11		fnelnfp 7181	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
12	11	notbid 321	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹‘𝑥) ≠ 𝑥))
13		nne 2964	. . . . . 6 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
14	12, 13	bitrdi 290	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) = 𝑥))
15	14	pm5.32da 590	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
16	3, 10, 15	3bitr4rd 315	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
17	1, 16	bitrid 286	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
18	17	eqrdv 2763	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906   I cid 5557  dom cdm 5663   Fn wfn 6535  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548

This theorem is used by:  symgcom2  33468