Theorem fxpgaeq 33134

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 7401	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝𝐴𝑋) = (𝑃𝐴𝑋))
2	1	eqeq1d 2732	. 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 33132	. . . . 5 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
7	3, 6	eleqtrd 2831	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
8		oveq2 7402	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	8, 9	eqeq12d 2746	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
11	10	ralbidv 3158	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
12	11	elrab 3667	. . . 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 3598	1 ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  {crab 3411  ‘cfv 6519  (class class class)co 7394  Basecbs 17185   GrpAct cga 19227  FixPtscfxp 33128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-map 8805  df-ga 19228  df-fxp 33129

This theorem is referenced by:  fxpsubm  33137