Theorem fxpgaval 33230

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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐶 = ∅)
2	1	rabeqdv 3415	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
3		rab0 4339	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅
4	2, 3	eqtrdi 2788	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6		gaset 19226	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
8	7, 5	fxpval 33228	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
10	1	rabeqdv 3415	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
11		rab0 4339	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅
12	10, 11	eqtrdi 2788	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅)
13	4, 9, 12	3eqtr4d 2782	. 2 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
14	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
15		fxpgaval.s	. . . . . . . . . 10 ⊢ 𝑈 = (Base‘𝐺)
16	15	gaf 19228	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐴:(𝑈 × 𝐶)⟶𝐶)
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(𝑈 × 𝐶)⟶𝐶)
18	17	fdmd 6673	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = (𝑈 × 𝐶))
19	18	dmeqd 5855	. . . . . 6 ⊢ (𝜑 → dom dom 𝐴 = dom (𝑈 × 𝐶))
20		dmxp 5879	. . . . . 6 ⊢ (𝐶 ≠ ∅ → dom (𝑈 × 𝐶) = 𝑈)
21	19, 20	sylan9eq 2792	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → dom dom 𝐴 = 𝑈)
22	21	raleqdv 3297	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥))
23	22	rabbidv 3407	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
24	14, 23	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
25	13, 24	pm2.61dane 3020	1 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3400  Vcvv 3441  ∅c0 4286   × cxp 5623  dom cdm 5625  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  Basecbs 17140   GrpAct cga 19222  FixPtscfxp 33226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-ga 19223  df-fxp 33227

This theorem is referenced by:  isfxp  33231  fxpgaeq  33232  cntrval2  33234