Theorem setrec1lem4 50501

Description: Lemma for setrec1 50502. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴).

In the proof of setrec1 50502, 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 1868, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 50499 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 23	. . . . . . . 8 ⊢ (𝑤 ⊆ 𝑋 → 𝑤 ⊆ 𝑋)
3		ssun1 4131	. . . . . . . 8 ⊢ 𝑋 ⊆ (𝑋 ∪ (𝐹‘𝐴))
4	2, 3	sstrdi 3950	. . . . . . 7 ⊢ (𝑤 ⊆ 𝑋 → 𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)))
5	4	imim1i 64	. . . . . 6 ⊢ ((𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)))
6	5	alimi 1844	. . . . 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 50498	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
10	7, 9	mpbid 235	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
11		sp 2222	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
13		setrec1lem4.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
14		sstr2 3945	. . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑋 → (𝑋 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
16	12, 15	syld 48	. . . . . . 7 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝐴 ⊆ 𝑧))
17		setrec1lem4.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V)
18		sseq1 3963	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
19		sseq1 3963	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑧))
20		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
21	20	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑤) ⊆ 𝑧 ↔ (𝐹‘𝐴) ⊆ 𝑧))
22	19, 21	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) ↔ (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧)))
23	18, 22	imbi12d 347	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → ((𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) ↔ (𝐴 ⊆ 𝑋 → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧))))
24	17, 23	spcdvw 50490	. . . . . . . 8 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐴 ⊆ 𝑋 → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧))))
25	13, 24	mpid 45	. . . . . . 7 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐴 ⊆ 𝑧 → (𝐹‘𝐴) ⊆ 𝑧)))
26	16, 25	mpdd 44	. . . . . 6 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝐹‘𝐴) ⊆ 𝑧))
27	12, 26	jcad 522	. . . . 5 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧)))
28	6, 27	syl5 35	. . . 4 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧)))
29		unss 4143	. . . 4 ⊢ ((𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)
30	28, 29	imbitrdi 254	. . 3 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
31	1, 30	alrimi 2252	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
32		fvex 6898	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
33		unexg 7747	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝐴) ∈ V) → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
34	7, 32, 33	sylancl 598	. . 3 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
35	8, 34	setrec1lem1 50498	. 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 260	1 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  {cab 2743  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  ‘cfv 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548

This theorem is used by:  setrec1  50502