Theorem elsetpreimafvb 48465

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 2853	. 2 ⊢ (𝑆 ∈ 𝑃 ↔ 𝑆 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})})
3		eqeq1 2765	. . . 4 ⊢ (𝑧 = 𝑆 → (𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
4	3	rexbidv 3187	. . 3 ⊢ (𝑧 = 𝑆 → (∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)}) ↔ ∃𝑥 ∈ 𝐴 𝑆 = (◡𝐹 “ {(𝐹‘𝑥)})))
5	4	elabg 3630	. 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)})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087  {csn 4584  ^◡ccnv 5650   “ cima 5654  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rex 3088

This theorem is used by:  elsetpreimafv  48466  preimafvelsetpreimafv  48469  0nelsetpreimafv  48471