Theorem setpreimafvex 48160

Description: The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
setpreimafvex	⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

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

Allowed substitution hints:   𝑃(𝑥, 𝑧)   𝑉(𝑥, 𝑧)

Proof of Theorem setpreimafvex

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. 2 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2		abrexexg 7956	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ∈ V)
3	1, 2	eqeltrid 2866	1 ⊢ (𝐴 ∈ 𝑉 → 𝑃 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {cab 2740  ∃wrex 3088  Vcvv 3454  {csn 4588  ^◡ccnv 5659   “ cima 5663  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-rep 5237

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456

This theorem is used by:  fundcmpsurbijinjpreimafv  48184