Theorem isfxp 33244

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 33243	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
5	4	eleq2d 2823	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥}))
6		oveq2 7368	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
7		id 22	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
8	6, 7	eqeq12d 2753	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
9	8	ralbidv 3161	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
10	9	elrab 3635	. . 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 708	1 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  ‘cfv 6492  (class class class)co 7360  Basecbs 17170   GrpAct cga 19255  FixPtscfxp 33239

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 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  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 7363  df-oprab 7364  df-mpo 7365  df-map 8768  df-ga 19256  df-fxp 33240

This theorem is referenced by:  fxpsubm  33248  fxpsubg  33249  fxpsubrg  33250  fxpsdrg  33251  issply  33720