Theorem setrec1lem4 49554

Description: Lemma for setrec1 49555. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴).

In the proof of setrec1 49555, 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 1835, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 49552 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 4153	. . . . . . . 8 ⊢ 𝑋 ⊆ (𝑋 ∪ (𝐹‘𝐴))
4	2, 3	sstrdi 3971	. . . . . . 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 1811	. . . . 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 49551	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
10	7, 9	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
11		sp 2183	. . . . . . 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 3965	. . . . . . . . 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 3984	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
19		sseq1 3984	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑧))
20		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
21	20	sseq1d 3990	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝑧))
22	19, 21	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧)))
23	18, 22	imbi12d 344	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝐴 ⊆ 𝑋 → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧))))
24	17, 23	spcdvw 49543	. . . . . . . 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 512	. . . . 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 4165	. . . 4 ⊢ ((𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)
30	28, 29	imbitrdi 251	. . 3 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
31	1, 30	alrimi 2213	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
32		fvex 6889	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
33		unexg 7737	. . . 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 49551	. 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 257	1 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  {cab 2713  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  ‘cfv 6531

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539

This theorem is referenced by:  setrec1  49555