Theorem fxpss 33259

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 33258	. 2 ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥})
4		ssrab2 4034	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥} ⊆ 𝐵
5	3, 4	eqsstrdi 3980	1 ⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401   ⊆ wss 3903  dom cdm 5632  (class class class)co 7368  FixPtscfxp 33256

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 5243  ax-pr 5379

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-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-fxp 33257

This theorem is referenced by:  fxpsubm  33265  fxpsubg  33266  fxpsubrg  33267  fxpsdrg  33268