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Theorem setrec1lem1 44097
 Description: Lemma for setrec1 44101. This is a utility theorem showing the equivalence of the statement 𝑋 ∈ 𝑌 and its expanded form. The proof uses elabg 3574 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 sseq2 3879 . . . . . . 7 (𝑦 = 𝑋 → (𝑤𝑦𝑤𝑋))
32imbi1d 334 . . . . . 6 (𝑦 = 𝑋 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
43albidv 1879 . . . . 5 (𝑦 = 𝑋 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
5 sseq1 3878 . . . . 5 (𝑦 = 𝑋 → (𝑦𝑧𝑋𝑧))
64, 5imbi12d 337 . . . 4 (𝑦 = 𝑋 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
76albidv 1879 . . 3 (𝑦 = 𝑋 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
8 setrec1lem1.1 . . 3 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
97, 8elab2g 3578 . 2 (𝑋𝑉 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
101, 9syl 17 1 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505   = wceq 1507   ∈ wcel 2048  {cab 2753   ⊆ wss 3825  ‘cfv 6182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-in 3832  df-ss 3839 This theorem is referenced by:  setrec1lem2  44098  setrec1lem4  44100  setrec2fun  44102
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