Theorem setrec1lem1 46393

Description: Lemma for setrec1 46397. This is a utility theorem showing the equivalence of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof uses elabg 3607 and equivalence theorems.

Variable 𝑌 is the class of sets 𝑦 that are recursively generated by the function 𝐹. In other words, 𝑦 ∈ 𝑌 iff by starting with the empty set and repeatedly applying 𝐹 to subsets 𝑤 of our set, we will eventually generate all the elements of 𝑌. In this theorem, 𝑋 is any element of 𝑌, and 𝑉 is any class. (Contributed by Emmett Weisz, 16-Oct-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
setrec1lem1.1	⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
setrec1lem1.2	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
setrec1lem1	⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))

Distinct variable groups:   𝑦,𝐹   𝑤,𝑋,𝑦   𝑧,𝑋,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑤)   𝐹(𝑧,𝑤)   𝑉(𝑦,𝑧,𝑤)   𝑌(𝑦,𝑧,𝑤)

Proof of Theorem setrec1lem1

Step	Hyp	Ref	Expression
1		setrec1lem1.2	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		sseq2 3947	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑤 ⊆ 𝑦 ↔ 𝑤 ⊆ 𝑋))
3	2	imbi1d 342	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))))
4	3	albidv 1923	. . . . 5 ⊢ (𝑦 = 𝑋 → (∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))))
5		sseq1 3946	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑧))
6	4, 5	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝑋 → ((∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
7	6	albidv 1923	. . 3 ⊢ (𝑦 = 𝑋 → (∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
8		setrec1lem1.1	. . 3 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
9	7, 8	elab2g 3611	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
10	1, 9	syl 17	1 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715   ⊆ wss 3887  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  setrec1lem2  46394  setrec1lem4  46396  setrec2fun  46398