Theorem isfxp 33250

Description: Property of being a fixed point. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpgaval.s	⊢ 𝑈 = (Base‘𝐺)
fxpgaval.a	⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
isfxp.x	⊢ (𝜑 → 𝑋 ∈ 𝐶)

Assertion

Ref	Expression
isfxp	⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))

Distinct variable groups:   𝐴,𝑝   𝐺,𝑝   𝑈,𝑝   𝑋,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝐶(𝑝)

Proof of Theorem isfxp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfxp.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐶)
2		fxpgaval.s	. . . . 5 ⊢ 𝑈 = (Base‘𝐺)
3		fxpgaval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
4	2, 3	fxpgaval 33249	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
5	4	eleq2d 2822	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥}))
6		oveq2 7366	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
7		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
8	6, 7	eqeq12d 2752	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
9	8	ralbidv 3159	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
10	9	elrab 3646	. . 3 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
11	5, 10	bitrdi 287	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)))
12	1, 11	mpbirand 707	1 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   GrpAct cga 19218  FixPtscfxp 33245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-ga 19219  df-fxp 33246

This theorem is referenced by:  fxpsubm  33254  fxpsubg  33255  fxpsubrg  33256  fxpsdrg  33257  issply  33719