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Theorem setrec1lem4 47575
Description: Lemma for setrec1 47576. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹𝐴).

In the proof of setrec1 47576, 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 1838, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 47573 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 4170 . . . . . . . 8 𝑋 ⊆ (𝑋 ∪ (𝐹𝐴))
42, 3sstrdi 3992 . . . . . . 7 (𝑤𝑋𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)))
54imim1i 63 . . . . . 6 ((𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
65alimi 1814 . . . . 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 47572 . . . . . . . 8 (𝜑 → (𝑋𝑌 ↔ ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧)))
107, 9mpbid 231 . . . . . . 7 (𝜑 → ∀𝑧(∀𝑤(𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑋𝑧))
11 sp 2177 . . . . . . 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 3987 . . . . . . . . 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 4005 . . . . . . . . . 10 (𝑤 = 𝐴 → (𝑤𝑋𝐴𝑋))
19 sseq1 4005 . . . . . . . . . . 11 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
20 fveq2 6881 . . . . . . . . . . . 12 (𝑤 = 𝐴 → (𝐹𝑤) = (𝐹𝐴))
2120sseq1d 4011 . . . . . . . . . . 11 (𝑤 = 𝐴 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐹𝐴) ⊆ 𝑧))
2219, 21imbi12d 345 . . . . . . . . . 10 (𝑤 = 𝐴 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧)))
2318, 22imbi12d 345 . . . . . . . . 9 (𝑤 = 𝐴 → ((𝑤𝑋 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝐴𝑋 → (𝐴𝑧 → (𝐹𝐴) ⊆ 𝑧))))
2417, 23spcdvw 47564 . . . . . . . 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 514 . . . . 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 4182 . . . 4 ((𝑋𝑧 ∧ (𝐹𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧)
3028, 29syl6ib 251 . . 3 (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
311, 30alrimi 2207 . 2 (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹𝐴)) → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹𝐴)) ⊆ 𝑧))
32 fvex 6894 . . . 4 (𝐹𝐴) ∈ V
33 unexg 7723 . . . 4 ((𝑋𝑌 ∧ (𝐹𝐴) ∈ V) → (𝑋 ∪ (𝐹𝐴)) ∈ V)
347, 32, 33sylancl 587 . . 3 (𝜑 → (𝑋 ∪ (𝐹𝐴)) ∈ V)
358, 34setrec1lem1 47572 . 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 397  wal 1540   = wceq 1542  wnf 1786  wcel 2107  {cab 2710  Vcvv 3475  cun 3944  wss 3946  cfv 6535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543
This theorem is referenced by:  setrec1  47576
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