Theorem elsetpreimafvb 48021

Description: The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)

Hypothesis

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

Assertion

Ref	Expression
elsetpreimafvb	⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))

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

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

Proof of Theorem elsetpreimafvb

Step	Hyp	Ref	Expression
1		setpreimafvex.p	. . 3 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
2	1	eleq2i 2861	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ 𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})})
3		eqeq1 2773	. . . 4 ⊢ (𝑧 = 𝑆 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
4	3	rexbidv 3195	. . 3 ⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
5	4	elabg 3644	. 2 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})} ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
6	2, 5	bitrid 286	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ 𝑃 ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {cab 2747  ∃wrex 3095  {csn 4594  ^◡ccnv 5661   “ cima 5665  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rex 3096

This theorem is referenced by:  elsetpreimafv  48022  preimafvelsetpreimafv  48025  0nelsetpreimafv  48027