Theorem nfpconfp 32724

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 3893	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2		fnelfp 7119	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
3	2	pm5.32da 584	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
4		inss1 4165	. . . . . . . 8 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
5		dmss 5844	. . . . . . . 8 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
7		fndm 6588	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	6, 7	sseqtrid 3957	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
9	8	sseld 3914	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) → 𝑥 ∈ 𝐴))
10	9	pm4.71rd 567	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I ))))
11		fnelnfp 7121	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
12	11	notbid 319	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹‘𝑥) ≠ 𝑥))
13		nne 2938	. . . . . 6 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
14	12, 13	bitrdi 288	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) = 𝑥))
15	14	pm5.32da 584	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
16	3, 10, 15	3bitr4rd 313	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
17	1, 16	bitrid 284	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
18	17	eqrdv 2737	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883   I cid 5512  dom cdm 5618   Fn wfn 6480  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493

This theorem is referenced by:  symgcom2  33165