Theorem fxpgaeq 33712

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 7419	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝𝐴𝑋) = (𝑃𝐴𝑋))
2	1	eqeq1d 2763	. 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 33710	. . . . 5 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
7	3, 6	eleqtrd 2863	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
8		oveq2 7420	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
9		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	8, 9	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
11	10	ralbidv 3186	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
12	11	elrab 3645	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
13	7, 12	sylib 221	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
14	13	simprd 501	. 2 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)
15		fxpgaeq.p	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑈)
16	2, 14, 15	rspcdva 3578	1 ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  ‘cfv 6531  (class class class)co 7412  Basecbs 17367   GrpAct cga 19483  FixPtscfxp 33706

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8833  df-ga 19484  df-fxp 33707

This theorem is used by:  fxpsubm  33715  fxpsubg  33716  fxpsubrg  33717  fxpsdrg  33718