Theorem fxpgaval 33798

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 3431	. . . 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 2814	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6		gaset 19537	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
7	5, 6	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
8	7, 5	fxpval 33796	. . . 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 3431	. . . 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 2814	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅)
13	4, 9, 12	3eqtr4d 2808	. 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 19539	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐴:(𝑈 × 𝐶)⟶𝐶)
17	5, 16	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(𝑈 × 𝐶)⟶𝐶)
18	17	fdmd 6742	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = (𝑈 × 𝐶))
19	18	dmeqd 5894	. . . . . 6 ⊢ (𝜑 → dom dom 𝐴 = dom (𝑈 × 𝐶))
20		dmxp 5918	. . . . . 6 ⊢ (𝐶 ≠ ∅ → dom (𝑈 × 𝐶) = 𝑈)
21	19, 20	sylan9eq 2818	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → dom dom 𝐴 = 𝑈)
22	21	raleqdv 3323	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥))
23	22	rabbidv 3423	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
24	14, 23	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
25	13, 24	pm2.61dane 3046	1 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  {crab 3416  Vcvv 3454  ∅c0 4281   × cxp 5653  dom cdm 5655  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443  Basecbs 17416   GrpAct cga 19533  FixPtscfxp 33794

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-map 8877  df-ga 19534  df-fxp 33795

This theorem is used by:  isfxp  33799  fxpgaeq  33800  cntrval2  33802