Theorem nfpconfp 32710

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 3911	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ dom (𝐹 ∖ I )) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ dom (𝐹 ∖ I )))
2		fnelfp 7121	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑥) = 𝑥))
3	2	pm5.32da 579	. . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom (𝐹 ∩ I )) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑥)))
4		inss1 4189	. . . . . . . 8 ⊢ (𝐹 ∩ I ) ⊆ 𝐹
5		dmss 5851	. . . . . . . 8 ⊢ ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ dom (𝐹 ∩ I ) ⊆ dom 𝐹
7		fndm 6595	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
8	6, 7	sseqtrid 3976	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) ⊆ 𝐴)
9	8	sseld 3932	. . . . 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 7123	. . . . . . 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 2936	. . . . . 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 579	. . . 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 2734	1 ⊢ (𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901   I cid 5518  dom cdm 5624   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  symgcom2  33166