Theorem fxpgaval 33609

Description: Value of the set of fixed points for a group action 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpgaval.s	⊢ 𝑈 = (Base‘𝐺)
fxpgaval.a	⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))

Assertion

Ref	Expression
fxpgaval	⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})

Distinct variable groups:   𝐴,𝑝,𝑥   𝑥,𝐶   𝐺,𝑝   𝑈,𝑝   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑝)   𝐶(𝑝)   𝑈(𝑥)   𝐺(𝑥)

Proof of Theorem fxpgaval

Step	Hyp	Ref	Expression
1		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐶 = ∅)
2	1	rabeqdv 3429	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
3		rab0 4338	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅
4	2, 3	eqtrdi 2813	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6		gaset 19424	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
7	5, 6	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
8	7, 5	fxpval 33607	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
10	1	rabeqdv 3429	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
11		rab0 4338	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅
12	10, 11	eqtrdi 2813	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅)
13	4, 9, 12	3eqtr4d 2807	. 2 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
14	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
15		fxpgaval.s	. . . . . . . . . 10 ⊢ 𝑈 = (Base‘𝐺)
16	15	gaf 19426	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐴:(𝑈 × 𝐶)⟶𝐶)
17	5, 16	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(𝑈 × 𝐶)⟶𝐶)
18	17	fdmd 6717	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = (𝑈 × 𝐶))
19	18	dmeqd 5893	. . . . . 6 ⊢ (𝜑 → dom dom 𝐴 = dom (𝑈 × 𝐶))
20		dmxp 5917	. . . . . 6 ⊢ (𝐶 ≠ ∅ → dom (𝑈 × 𝐶) = 𝑈)
21	19, 20	sylan9eq 2817	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → dom dom 𝐴 = 𝑈)
22	21	raleqdv 3321	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥))
23	22	rabbidv 3421	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
24	14, 23	eqtrd 2797	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
25	13, 24	pm2.61dane 3044	1 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  {crab 3414  Vcvv 3453  ∅c0 4282   × cxp 5657  dom cdm 5659  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  Basecbs 17305   GrpAct cga 19420  FixPtscfxp 33605

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8831  df-ga 19421  df-fxp 33606

This theorem is used by:  isfxp  33610  fxpgaeq  33611  cntrval2  33613