Theorem fxpgaeq 33806

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 7453	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝𝐴𝑋) = (𝑃𝐴𝑋))
2	1	eqeq1d 2765	. 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 33804	. . . . 5 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
7	3, 6	eleqtrd 2866	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
8		oveq2 7454	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
9		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	8, 9	eqeq12d 2779	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
11	10	ralbidv 3189	. . . . 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 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 3581	1 ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  ‘cfv 6561  (class class class)co 7446  Basecbs 17419   GrpAct cga 19536  FixPtscfxp 33800

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 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-map 8880  df-ga 19537  df-fxp 33801

This theorem is used by:  fxpsubm  33809  fxpsubg  33810  fxpsubrg  33811  fxpsdrg  33812