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Theorem setrec1lem1 42233
 Description: Lemma for setrec1 42237. This is a utility theorem showing the equivalence of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof uses elabg 3320 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(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
setrec1lem1.2 (𝜑𝑋𝑉)
Assertion
Ref Expression
setrec1lem1 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
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
StepHypRef Expression
1 setrec1lem1.2 . 2 (𝜑𝑋𝑉)
2 setrec1lem1.1 . . . 4 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
32eleq2i 2680 . . 3 (𝑋𝑌𝑋 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
4 sseq2 3590 . . . . . . . 8 (𝑦 = 𝑋 → (𝑤𝑦𝑤𝑋))
54imbi1d 330 . . . . . . 7 (𝑦 = 𝑋 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
65albidv 1836 . . . . . 6 (𝑦 = 𝑋 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
7 sseq1 3589 . . . . . 6 (𝑦 = 𝑋 → (𝑦𝑧𝑋𝑧))
86, 7imbi12d 333 . . . . 5 (𝑦 = 𝑋 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
98albidv 1836 . . . 4 (𝑦 = 𝑋 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
109elabg 3320 . . 3 (𝑋𝑉 → (𝑋 ∈ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
113, 10syl5bb 271 . 2 (𝑋𝑉 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
121, 11syl 17 1 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596   ⊆ wss 3540  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  setrec1lem2  42234  setrec1lem4  42236  setrec2fun  42238
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