Theorem isfxp 33608

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 33607	. . . 4 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
5	4	eleq2d 2846	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥}))
6		oveq2 7421	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
7		id 23	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
8	6, 7	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
9	8	ralbidv 3185	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
10	9	elrab 3645	. . 3 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
11	5, 10	bitrdi 290	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)))
12	1, 11	mpbirand 720	1 ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  ‘cfv 6533  (class class class)co 7413  Basecbs 17301   GrpAct cga 19416  FixPtscfxp 33603

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8828  df-ga 19417  df-fxp 33604

This theorem is used by:  fxpsubm  33612  fxpsubg  33613  fxpsubrg  33614  fxpsdrg  33615  issply  34071