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Theorem setrec1lem2 42200
 Description: Lemma for setrec1 42203. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3585, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)
Hypotheses
Ref Expression
setrec1lem2.1 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
setrec1lem2.2 (𝜑𝑋𝑉)
setrec1lem2.3 (𝜑𝑋𝑌)
Assertion
Ref Expression
setrec1lem2 (𝜑 𝑋𝑌)
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 setrec1lem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 setrec1lem2.3 . . . . . . 7 (𝜑𝑋𝑌)
2 dfss3 3585 . . . . . . 7 (𝑋𝑌 ↔ ∀𝑥𝑋 𝑥𝑌)
31, 2sylib 208 . . . . . 6 (𝜑 → ∀𝑥𝑋 𝑥𝑌)
4 setrec1lem2.1 . . . . . . . 8 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
5 vex 3198 . . . . . . . . 9 𝑥 ∈ V
65a1i 11 . . . . . . . 8 (𝜑𝑥 ∈ V)
74, 6setrec1lem1 42199 . . . . . . 7 (𝜑 → (𝑥𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
87ralbidv 2983 . . . . . 6 (𝜑 → (∀𝑥𝑋 𝑥𝑌 ↔ ∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
93, 8mpbid 222 . . . . 5 (𝜑 → ∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
10 ralcom4 3219 . . . . 5 (∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) ↔ ∀𝑧𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
119, 10sylib 208 . . . 4 (𝜑 → ∀𝑧𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
12 nfra1 2938 . . . . . 6 𝑥𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)
13 nfv 1841 . . . . . 6 𝑥𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
14 rsp 2926 . . . . . . . 8 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (𝑥𝑋 → (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
15 elssuni 4458 . . . . . . . . . . . 12 (𝑥𝑋𝑥 𝑋)
16 sstr2 3602 . . . . . . . . . . . 12 (𝑤𝑥 → (𝑥 𝑋𝑤 𝑋))
1715, 16syl5com 31 . . . . . . . . . . 11 (𝑥𝑋 → (𝑤𝑥𝑤 𝑋))
1817imim1d 82 . . . . . . . . . 10 (𝑥𝑋 → ((𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
1918alimdv 1843 . . . . . . . . 9 (𝑥𝑋 → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))))
2019imim1d 82 . . . . . . . 8 (𝑥𝑋 → ((∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
2114, 20sylcom 30 . . . . . . 7 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (𝑥𝑋 → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
2221com23 86 . . . . . 6 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑥𝑋𝑥𝑧)))
2312, 13, 22ralrimd 2956 . . . . 5 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2423alimi 1737 . . . 4 (∀𝑧𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2511, 24syl 17 . . 3 (𝜑 → ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
26 unissb 4460 . . . . 5 ( 𝑋𝑧 ↔ ∀𝑥𝑋 𝑥𝑧)
2726imbi2i 326 . . . 4 ((∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) ↔ (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2827albii 1745 . . 3 (∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) ↔ ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2925, 28sylibr 224 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
30 setrec1lem2.2 . . . 4 (𝜑𝑋𝑉)
31 uniexg 6940 . . . 4 (𝑋𝑉 𝑋 ∈ V)
3230, 31syl 17 . . 3 (𝜑 𝑋 ∈ V)
334, 32setrec1lem1 42199 . 2 (𝜑 → ( 𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
3429, 33mpbird 247 1 (𝜑 𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479   = wceq 1481   ∈ wcel 1988  {cab 2606  ∀wral 2909  Vcvv 3195   ⊆ wss 3567  ∪ cuni 4427  ‘cfv 5876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428 This theorem is referenced by:  setrec1lem3  42201
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