Theorem setrec1lem4 50180

Description: Lemma for setrec1 50181. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴).

In the proof of setrec1 50181, 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 1842, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 50178 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 4107	. . . . . . . 8 ⊢ 𝑋 ⊆ (𝑋 ∪ (𝐹‘𝐴))
4	2, 3	sstrdi 3927	. . . . . . 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 1818	. . . . 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 50177	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧)))
10	7, 9	mpbid 233	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑋 ⊆ 𝑧))
11		sp 2195	. . . . . . 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 3922	. . . . . . . . 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 3940	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
19		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑧))
20		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
21	20	sseq1d 3946	. . . . . . . . . . 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 50169	. . . . . . . 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 517	. . . . 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 4119	. . . 4 ⊢ ((𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)
30	28, 29	imbitrdi 252	. . 3 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
31	1, 30	alrimi 2225	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
32		fvex 6840	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
33		unexg 7686	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝐴) ∈ V) → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
34	7, 32, 33	sylancl 592	. . 3 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
35	8, 34	setrec1lem1 50177	. 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 258	1 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ 𝑌)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  {cab 2717  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493

This theorem is referenced by:  setrec1  50181