Theorem preimafvsspwdm 45734

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 45731	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑠 ∈ 𝑃) → 𝑠 ⊆ 𝐴)
3	2	ralrimiva 3145	. 2 ⊢ (𝐹 Fn 𝐴 → ∀𝑠 ∈ 𝑃 𝑠 ⊆ 𝐴)
4		pwssb 5081	. 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 1541  {cab 2708  ∀wral 3060  ∃wrex 3069   ⊆ wss 3928  𝒫 cpw 4580  {csn 4606  ^◡ccnv 5652   “ cima 5656   Fn wfn 6511  ‘cfv 6516

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-xp 5659  df-cnv 5661  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-fn 6519

This theorem is referenced by: (None)