Theorem setrec1lem4 46396

Description: Lemma for setrec1 46397. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴).

In the proof of setrec1 46397, the following is substituted for this theorem's 𝜑: (𝜑 ∧ (𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ {𝑦 ∣ ∀𝑧(∀𝑤 (𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)})) 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 Ⅎ𝑧𝜑, but the proof would be harder to read because theorems in deduction form would be interrupted by theorems like eximi 1837, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 46394 could similarly be made easier to read by adding the hypothesis Ⅎ𝑧𝜑, 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	⊢ Ⅎ𝑧𝜑
setrec1lem4.2	⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
setrec1lem4.3	⊢ (𝜑 → 𝐴 ∈ V)
setrec1lem4.4	⊢ (𝜑 → 𝐴 ⊆ 𝑋)
setrec1lem4.5	⊢ (𝜑 → 𝑋 ∈ 𝑌)

Assertion

Ref	Expression
setrec1lem4	⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 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

Step	Hyp	Ref	Expression
1		setrec1lem4.1	. . 3 ⊢ Ⅎ𝑧𝜑
2		id 22	. . . . . . . 8 ⊢ (𝑤 ⊆ 𝑋 → 𝑤 ⊆ 𝑋)
3		ssun1 4106	. . . . . . . 8 ⊢ 𝑋 ⊆ (𝑋 ∪ (𝐹‘𝐴))
4	2, 3	sstrdi 3933	. . . . . . 7 ⊢ (𝑤 ⊆ 𝑋 → 𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)))
5	4	imim1i 63	. . . . . 6 ⊢ ((𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
6	5	alimi 1814	. . . . 5 ⊢ (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
7		setrec1lem4.5	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
8		setrec1lem4.2	. . . . . . . . 9 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
9	8, 7	setrec1lem1 46393	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
10	7, 9	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
11		sp 2176	. . . . . . 7 ⊢ (∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧) → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
13		setrec1lem4.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
14		sstr2 3928	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑋 → (𝑋 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
16	12, 15	syld 47	. . . . . . 7 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝐴 ⊆ 𝑧))
17		setrec1lem4.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
18		sseq1 3946	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
19		sseq1 3946	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑧))
20		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
21	20	sseq1d 3952	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝑧))
22	19, 21	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧)))
23	18, 22	imbi12d 345	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝐴 ⊆ 𝑋 → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧))))
24	17, 23	spcdvw 46385	. . . . . . . 8 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐴 ⊆ 𝑋 → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧))))
25	13, 24	mpid 44	. . . . . . 7 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧)))
26	16, 25	mpdd 43	. . . . . 6 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐹‘𝐴) ⊆ 𝑧))
27	12, 26	jcad 513	. . . . 5 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧)))
28	6, 27	syl5 34	. . . 4 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧)))
29		unss 4118	. . . 4 ⊢ ((𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)
30	28, 29	syl6ib 250	. . 3 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
31	1, 30	alrimi 2206	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
32		fvex 6787	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
33		unexg 7599	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝐴) ∈ V) → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
34	7, 32, 33	sylancl 586	. . 3 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
35	8, 34	setrec1lem1 46393	. 2 ⊢ (𝜑 → ((𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)))
36	31, 35	mpbird 256	1 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  {cab 2715  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  ‘cfv 6433

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  setrec1  46397