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Theorem setrec1lem4 47822
Description: Lemma for setrec1 47823. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 47823, 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 1835, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 47820 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 4171 . . . . . . . 8 𝑋 ⊆ (𝑋 ∪ (𝐹𝐴))
42, 3sstrdi 3993 . . . . . . 7 (𝑤𝑋𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)))
54imim1i 63 . . . . . 6 ((𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
65alimi 1811 . . . . 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 47819 . . . . . . . 8 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
107, 9mpbid 231 . . . . . . 7 (𝜑 → ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
11 sp 2174 . . . . . . 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 3988 . . . . . . . . 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 4006 . . . . . . . . . 10 (𝑤 = 𝐴 → (𝑤𝑋𝐴𝑋))
19 sseq1 4006 . . . . . . . . . . 11 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
20 fveq2 6890 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝐹𝑤) = (𝐹𝐴))
2120sseq1d 4012 . . . . . . . . . . 11 (𝑤 = 𝐴 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐹𝐴) ⊆ 𝑧))
2219, 21imbi12d 343 . . . . . . . . . 10 (𝑤 = 𝐴 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧)))
2318, 22imbi12d 343 . . . . . . . . 9 (𝑤 = 𝐴 → ((𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝐴𝑋 → (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧))))
2417, 23spcdvw 47811 . . . . . . . 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 511 . . . . 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 4183 . . . 4 ((𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)
3028, 29imbitrdi 250 . . 3 (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
311, 30alrimi 2204 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
32 fvex 6903 . . . 4 (𝐹𝐴) ∈ V
33 unexg 7738 . . . 4 ((𝑋𝑌 ∧ (𝐹𝐴) ∈ V) → (𝑋 ∪ (𝐹𝐴)) ∈ V)
347, 32, 33sylancl 584 . . 3 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ V)
358, 34setrec1lem1 47819 . 2 (𝜑 → ((𝑋 ∪ (𝐹𝐴)) ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)))
3631, 35mpbird 256 1 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1537   = wceq 1539  wnf 1783  wcel 2104  {cab 2707  Vcvv 3472  cun 3945  wss 3947  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550
This theorem is referenced by:  setrec1  47823
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