Theorem preimafvsspwdm 44729

Description: The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.)

Hypothesis

Ref	Expression
setpreimafvex.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}

Assertion

Ref	Expression
preimafvsspwdm	⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑃

Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem preimafvsspwdm

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	elsetpreimafvssdm 44726	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → 𝑠 ⊆ 𝐴)
3	2	ralrimiva 3107	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
4		pwssb 5026	. 2 ⊢ (𝑃 ⊆ 𝒫 𝐴 ↔ ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
5	3, 4	sylibr 233	1 ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1539  {cab 2715  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ^◡ccnv 5579   “ cima 5583   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-fn 6421

This theorem is referenced by: (None)