Theorem preimafvsspwdm 47314

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 47311	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → 𝑠 ⊆ 𝐴)
3	2	ralrimiva 3144	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
4		pwssb 5106	. 2 ⊢ (𝑃 ⊆ 𝒫 𝐴 ↔ ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
5	3, 4	sylibr 234	1 ⊢ (𝐹 Fn 𝐴 → 𝑃 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1537  {cab 2712  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  𝒫 cpw 4605  {csn 4631  ^◡ccnv 5688   “ cima 5692   Fn wfn 6558  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fn 6566

This theorem is referenced by: (None)