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Theorem setrec1lem2 45232
 Description: Lemma for setrec1 45235. 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 3903, 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 3903 . . . . . . 7 (𝑋𝑌 ↔ ∀𝑥𝑋 𝑥𝑌)
31, 2sylib 221 . . . . . 6 (𝜑 → ∀𝑥𝑋 𝑥𝑌)
4 setrec1lem2.1 . . . . . . . 8 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
5 vex 3444 . . . . . . . . 9 𝑥 ∈ V
65a1i 11 . . . . . . . 8 (𝜑𝑥 ∈ V)
74, 6setrec1lem1 45231 . . . . . . 7 (𝜑 → (𝑥𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
87ralbidv 3162 . . . . . 6 (𝜑 → (∀𝑥𝑋 𝑥𝑌 ↔ ∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
93, 8mpbid 235 . . . . 5 (𝜑 → ∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
10 ralcom4 3198 . . . . 5 (∀𝑥𝑋𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) ↔ ∀𝑧𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
119, 10sylib 221 . . . 4 (𝜑 → ∀𝑧𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧))
12 nfra1 3183 . . . . . 6 𝑥𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)
13 nfv 1915 . . . . . 6 𝑥𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
14 rsp 3170 . . . . . . . 8 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (𝑥𝑋 → (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
15 elssuni 4830 . . . . . . . . . . . 12 (𝑥𝑋𝑥 𝑋)
16 sstr2 3922 . . . . . . . . . . . 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 1917 . . . . . . . . 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 3182 . . . . 5 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2423alimi 1813 . . . 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 4832 . . . . 5 ( 𝑋𝑧 ↔ ∀𝑥𝑋 𝑥𝑧)
2726imbi2i 339 . . . 4 ((∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) ↔ (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2827albii 1821 . . 3 (∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) ↔ ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2925, 28sylibr 237 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
30 setrec1lem2.2 . . . 4 (𝜑𝑋𝑉)
3130uniexd 7450 . . 3 (𝜑 𝑋 ∈ V)
324, 31setrec1lem1 45231 . 2 (𝜑 → ( 𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
3329, 32mpbird 260 1 (𝜑 𝑋𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801 This theorem is referenced by:  setrec1lem3  45233
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