Theorem nfpconfp 32651

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 3986	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2		fnelfp 7209	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
3	2	pm5.32da 578	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
4		inss1 4258	. . . . . . . 8 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
5		dmss 5927	. . . . . . . 8 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
7		fndm 6682	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	6, 7	sseqtrid 4061	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
9	8	sseld 4007	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) → 𝑥 ∈ 𝐴))
10	9	pm4.71rd 562	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I ))))
11		fnelnfp 7211	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) ≠ 𝑥))
12	11	notbid 318	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ ¬ (𝐹‘𝑥) ≠ 𝑥))
13		nne 2950	. . . . . 6 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
14	12, 13	bitrdi 287	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑥) = 𝑥))
15	14	pm5.32da 578	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
16	3, 10, 15	3bitr4rd 312	. . 3 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
17	1, 16	bitrid 283	. 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ 𝑥 ∈ dom (𝐹 ∩ I )))
18	17	eqrdv 2738	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976   I cid 5592  dom cdm 5700   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  symgcom2  33077