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Theorem setrec1lem4 48921
Description: Lemma for setrec1 48922. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 48922, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴𝑥𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)})) Therefore, we cannot declare 𝑧 to be a distinct variable from 𝜑, since we need it to appear as a bound variable in 𝜑. This theorem can be proven without the hypothesis 𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1832, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 48919 could similarly be made easier to read by adding the hypothesis 𝑧𝜑, but I had already finished the proof and decided to leave it as is. (Contributed by Emmett Weisz, 26-Nov-2020.) (New usage is discouraged.)

Hypotheses
Ref Expression
setrec1lem4.1 𝑧𝜑
setrec1lem4.2 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
setrec1lem4.3 (𝜑𝐴 ∈ V)
setrec1lem4.4 (𝜑𝐴𝑋)
setrec1lem4.5 (𝜑𝑋𝑌)
Assertion
Ref Expression
setrec1lem4 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)
Distinct variable groups:   𝑦,𝑤,𝑧,𝐴   𝑤,𝐹,𝑦,𝑧   𝑤,𝑋,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑤)   𝑌(𝑦,𝑧,𝑤)

Proof of Theorem setrec1lem4
StepHypRef Expression
1 setrec1lem4.1 . . 3 𝑧𝜑
2 id 22 . . . . . . . 8 (𝑤𝑋𝑤𝑋)
3 ssun1 4188 . . . . . . . 8 𝑋 ⊆ (𝑋 ∪ (𝐹𝐴))
42, 3sstrdi 4008 . . . . . . 7 (𝑤𝑋𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)))
54imim1i 63 . . . . . 6 ((𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
65alimi 1808 . . . . 5 (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → ∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
7 setrec1lem4.5 . . . . . . . 8 (𝜑𝑋𝑌)
8 setrec1lem4.2 . . . . . . . . 9 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
98, 7setrec1lem1 48918 . . . . . . . 8 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
107, 9mpbid 232 . . . . . . 7 (𝜑 → ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
11 sp 2181 . . . . . . 7 (∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧) → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
1210, 11syl 17 . . . . . 6 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
13 setrec1lem4.4 . . . . . . . . 9 (𝜑𝐴𝑋)
14 sstr2 4002 . . . . . . . . 9 (𝐴𝑋 → (𝑋𝑧𝐴𝑧))
1513, 14syl 17 . . . . . . . 8 (𝜑 → (𝑋𝑧𝐴𝑧))
1612, 15syld 47 . . . . . . 7 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝐴𝑧))
17 setrec1lem4.3 . . . . . . . . 9 (𝜑𝐴 ∈ V)
18 sseq1 4021 . . . . . . . . . 10 (𝑤 = 𝐴 → (𝑤𝑋𝐴𝑋))
19 sseq1 4021 . . . . . . . . . . 11 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
20 fveq2 6907 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝐹𝑤) = (𝐹𝐴))
2120sseq1d 4027 . . . . . . . . . . 11 (𝑤 = 𝐴 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐹𝐴) ⊆ 𝑧))
2219, 21imbi12d 344 . . . . . . . . . 10 (𝑤 = 𝐴 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧)))
2318, 22imbi12d 344 . . . . . . . . 9 (𝑤 = 𝐴 → ((𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝐴𝑋 → (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧))))
2417, 23spcdvw 48910 . . . . . . . 8 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝐴𝑋 → (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧))))
2513, 24mpid 44 . . . . . . 7 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧)))
2616, 25mpdd 43 . . . . . 6 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝐹𝐴) ⊆ 𝑧))
2712, 26jcad 512 . . . . 5 (𝜑 → (∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧)))
286, 27syl5 34 . . . 4 (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧)))
29 unss 4200 . . . 4 ((𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)
3028, 29imbitrdi 251 . . 3 (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
311, 30alrimi 2211 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
32 fvex 6920 . . . 4 (𝐹𝐴) ∈ V
33 unexg 7762 . . . 4 ((𝑋𝑌 ∧ (𝐹𝐴) ∈ V) → (𝑋 ∪ (𝐹𝐴)) ∈ V)
347, 32, 33sylancl 586 . . 3 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ V)
358, 34setrec1lem1 48918 . 2 (𝜑 → ((𝑋 ∪ (𝐹𝐴)) ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)))
3631, 35mpbird 257 1 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wnf 1780  wcel 2106  {cab 2712  Vcvv 3478  cun 3961  wss 3963  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571
This theorem is referenced by:  setrec1  48922
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