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Theorem setrec1lem2 42760
Description: Lemma for setrec1 42763. 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 3625, 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 3625 . . . . . . 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 3234 . . . . . . . . 9 𝑥 ∈ V
65a1i 11 . . . . . . . 8 (𝜑𝑥 ∈ V)
74, 6setrec1lem1 42759 . . . . . . 7 (𝜑 → (𝑥𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
87ralbidv 3015 . . . . . 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 3255 . . . . 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 2970 . . . . . 6 𝑥𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)
13 nfv 1883 . . . . . 6 𝑥𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
14 rsp 2958 . . . . . . . 8 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (𝑥𝑋 → (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧)))
15 elssuni 4499 . . . . . . . . . . . 12 (𝑥𝑋𝑥 𝑋)
16 sstr2 3643 . . . . . . . . . . . 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 1885 . . . . . . . . 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 2988 . . . . 5 (∀𝑥𝑋 (∀𝑤(𝑤𝑥 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑥𝑧) → (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2423alimi 1779 . . . 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 4501 . . . . 5 ( 𝑋𝑧 ↔ ∀𝑥𝑋 𝑥𝑧)
2726imbi2i 325 . . . 4 ((∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) ↔ (∀𝑤(𝑤 𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑥𝑋 𝑥𝑧))
2827albii 1787 . . 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 6997 . . . 4 (𝑋𝑉 𝑋 ∈ V)
3230, 31syl 17 . . 3 (𝜑 𝑋 ∈ V)
334, 32setrec1lem1 42759 . 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 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  wss 3607   cuni 4468  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469
This theorem is referenced by:  setrec1lem3  42761
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