Theorem setrec1lem4 47735

Description: Lemma for setrec1 47736. If 𝑋 is recursively generated by 𝐹, then so is 𝑋 ∪ (𝐹‘𝐴).

In the proof of setrec1 47736, 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 1838, making the antecedent of each line something more complicated than 𝜑. The proof of setrec1lem2 47733 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 4173	. . . . . . . 8 ⊢ 𝑋 ⊆ (𝑋 ∪ (𝐹‘𝐴))
4	2, 3	sstrdi 3995	. . . . . . 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 47732	. . . . . . . 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 2177	. . . . . . 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 3990	. . . . . . . . 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 4008	. . . . . . . . . 10 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
19		sseq1 4008	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐴 → (𝑤 ⊆ 𝑧 ↔ 𝐴 ⊆ 𝑧))
20		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴))
21	20	sseq1d 4014	. . . . . . . . . . 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 47724	. . . . . . . 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 514	. . . . 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 4185	. . . 4 ⊢ ((𝑋 ⊆ 𝑧 ∧ (𝐹‘𝐴) ⊆ 𝑧) ↔ (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧)
30	28, 29	imbitrdi 250	. . 3 ⊢ (𝜑 → (∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
31	1, 30	alrimi 2207	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ (𝑋 ∪ (𝐹‘𝐴)) → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑋 ∪ (𝐹‘𝐴)) ⊆ 𝑧))
32		fvex 6905	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
33		unexg 7736	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝐴) ∈ V) → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
34	7, 32, 33	sylancl 587	. . 3 ⊢ (𝜑 → (𝑋 ∪ (𝐹‘𝐴)) ∈ V)
35	8, 34	setrec1lem1 47732	. 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 397  ∀wal 1540   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  {cab 2710  Vcvv 3475   ∪ cun 3947   ⊆ wss 3949  ‘cfv 6544

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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

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 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552

This theorem is referenced by:  setrec1  47736