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Theorem setrec1lem4 49209
Description: Lemma for setrec1 49210. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 49210, 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 49207 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 4178 . . . . . . . 8 𝑋 ⊆ (𝑋 ∪ (𝐹𝐴))
42, 3sstrdi 3996 . . . . . . 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 49206 . . . . . . . 8 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
107, 9mpbid 232 . . . . . . 7 (𝜑 → ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
11 sp 2183 . . . . . . 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 3990 . . . . . . . . 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 4009 . . . . . . . . . 10 (𝑤 = 𝐴 → (𝑤𝑋𝐴𝑋))
19 sseq1 4009 . . . . . . . . . . 11 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
20 fveq2 6906 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝐹𝑤) = (𝐹𝐴))
2120sseq1d 4015 . . . . . . . . . . 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 49198 . . . . . . . 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 4190 . . . 4 ((𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)
3028, 29imbitrdi 251 . . 3 (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
311, 30alrimi 2213 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
32 fvex 6919 . . . 4 (𝐹𝐴) ∈ V
33 unexg 7763 . . . 4 ((𝑋𝑌 ∧ (𝐹𝐴) ∈ V) → (𝑋 ∪ (𝐹𝐴)) ∈ V)
347, 32, 33sylancl 586 . . 3 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ V)
358, 34setrec1lem1 49206 . 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 1538   = wceq 1540  wnf 1783  wcel 2108  {cab 2714  Vcvv 3480  cun 3949  wss 3951  cfv 6561
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  setrec1  49210
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