Theorem fxpgaval 33255

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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐶 = ∅)
2	1	rabeqdv 3407	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
3		rab0 4321	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅
4	2, 3	eqtrdi 2791	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = ∅)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6		gaset 19266	. . . . . 6 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐶 ∈ V)
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ V)
8	7, 5	fxpval 33253	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
9	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
10	1	rabeqdv 3407	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
11		rab0 4321	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅
12	10, 11	eqtrdi 2791	. . 3 ⊢ ((𝜑 ∧ 𝐶 = ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} = ∅)
13	4, 9, 12	3eqtr4d 2785	. 2 ⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
14	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
15		fxpgaval.s	. . . . . . . . . 10 ⊢ 𝑈 = (Base‘𝐺)
16	15	gaf 19268	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐺 GrpAct 𝐶) → 𝐴:(𝑈 × 𝐶)⟶𝐶)
17	5, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:(𝑈 × 𝐶)⟶𝐶)
18	17	fdmd 6672	. . . . . . 7 ⊢ (𝜑 → dom 𝐴 = (𝑈 × 𝐶))
19	18	dmeqd 5854	. . . . . 6 ⊢ (𝜑 → dom dom 𝐴 = dom (𝑈 × 𝐶))
20		dmxp 5878	. . . . . 6 ⊢ (𝐶 ≠ ∅ → dom (𝑈 × 𝐶) = 𝑈)
21	19, 20	sylan9eq 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → dom dom 𝐴 = 𝑈)
22	21	raleqdv 3298	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥))
23	22	rabbidv 3399	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
24	14, 23	eqtrd 2775	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ ∅) → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
25	13, 24	pm2.61dane 3022	1 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  {crab 3392  Vcvv 3432  ∅c0 4268   × cxp 5623  dom cdm 5625  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   GrpAct cga 19262  FixPtscfxp 33251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-ga 19263  df-fxp 33252

This theorem is referenced by:  isfxp  33256  fxpgaeq  33257  cntrval2  33259