Theorem fxpss 33517

Description: The set of fixed points is a subset of the set acted upon. (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpval.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fxpval.2	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
fxpss	⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)

Proof of Theorem fxpss

Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fxpval.1	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
2		fxpval.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
3	1, 2	fxpval 33516	. 2 ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
4		ssrab2 4037	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} ⊆ 𝐵
5	3, 4	eqsstrdi 3984	1 ⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419   ⊆ wss 3908  dom cdm 5666  (class class class)co 7423  FixPtscfxp 33514

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-fxp 33515

This theorem is used by:  fxpsubm  33523  fxpsubg  33524  fxpsubrg  33525  fxpsdrg  33526