Theorem fxpgaeq 33262

Description: A fixed point 𝑋 is invariant under 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))
fxpgaeq.x	⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))
fxpgaeq.p	⊢ (𝜑 → 𝑃 ∈ 𝑈)

Assertion

Ref	Expression
fxpgaeq	⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

Proof of Theorem fxpgaeq

Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7375	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝𝐴𝑋) = (𝑃𝐴𝑋))
2	1	eqeq1d 2739	. 2 ⊢ (𝑝 = 𝑃 → ((𝑝𝐴𝑋) = 𝑋 ↔ (𝑃𝐴𝑋) = 𝑋))
3		fxpgaeq.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))
4		fxpgaval.s	. . . . . 6 ⊢ 𝑈 = (Base‘𝐺)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6	4, 5	fxpgaval 33260	. . . . 5 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
7	3, 6	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
8		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	8, 9	eqeq12d 2753	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
11	10	ralbidv 3161	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
12	11	elrab 3648	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
13	7, 12	sylib 218	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
14	13	simprd 495	. 2 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)
15		fxpgaeq.p	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑈)
16	2, 14, 15	rspcdva 3579	1 ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   GrpAct cga 19230  FixPtscfxp 33256

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-ga 19231  df-fxp 33257

This theorem is referenced by:  fxpsubm  33265  fxpsubg  33266  fxpsubrg  33267  fxpsdrg  33268