Theorem fxpss 33709

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 33708	. 2 ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
4		ssrab2 4028	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} ⊆ 𝐵
5	3, 4	eqsstrdi 3975	1 ⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413   ⊆ wss 3899  dom cdm 5651  (class class class)co 7412  FixPtscfxp 33706

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 2733  ax-sep 5249  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fxp 33707

This theorem is used by:  fxpsubm  33715  fxpsubg  33716  fxpsubrg  33717  fxpsdrg  33718