Theorem setrec1lem2 44785

Description: Lemma for setrec1 44788. If a family of sets are all recursively generated by 𝐹, so is their union. In this theorem, 𝑋 is a family of sets which are all elements of 𝑌, and 𝑉 is any class. Use dfss3 3955, equivalence and equality theorems, and unissb at the end. Sandwich with applications of setrec1lem1. (Contributed by Emmett Weisz, 24-Jan-2021.) (New usage is discouraged.)

Hypotheses

Ref	Expression
setrec1lem2.1	⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
setrec1lem2.2	⊢ (𝜑 → 𝑋 ∈ 𝑉)
setrec1lem2.3	⊢ (𝜑 → 𝑋 ⊆ 𝑌)

Assertion

Ref	Expression
setrec1lem2	⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌)

Distinct variable groups:   𝑦,𝐹   𝑤,𝑋,𝑦   𝑧,𝑋,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑤)   𝐹(𝑧,𝑤)   𝑉(𝑦,𝑧,𝑤)   𝑌(𝑦,𝑧,𝑤)

Proof of Theorem setrec1lem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setrec1lem2.3	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
2		dfss3 3955	. . . . . . 7 ⊢ (𝑋 ⊆ 𝑌 ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ 𝑌)
3	1, 2	sylib 220	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝑥 ∈ 𝑌)
4		setrec1lem2.1	. . . . . . . 8 ⊢ 𝑌 = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
5		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑥 ∈ V)
7	4, 6	setrec1lem1 44784	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
8	7	ralbidv 3197	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 𝑥 ∈ 𝑌 ↔ ∀𝑥 ∈ 𝑋 ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
9	3, 8	mpbid 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧))
10		ralcom4 3235	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑧(∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) ↔ ∀𝑧∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧))
11	9, 10	sylib 220	. . . 4 ⊢ (𝜑 → ∀𝑧∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧))
12		nfra1 3219	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)
13		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
14		rsp 3205	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (𝑥 ∈ 𝑋 → (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
15		elssuni 4860	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → 𝑥 ⊆ ∪ 𝑋)
16		sstr2 3973	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ 𝑥 → (𝑥 ⊆ ∪ 𝑋 → 𝑤 ⊆ ∪ 𝑋))
17	15, 16	syl5com 31	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (𝑤 ⊆ 𝑥 → 𝑤 ⊆ ∪ 𝑋))
18	17	imim1d 82	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋 → ((𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))))
19	18	alimdv 1913	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 → (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))))
20	19	imim1d 82	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
21	14, 20	sylcom 30	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (𝑥 ∈ 𝑋 → (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧)))
22	21	com23 86	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → (𝑥 ∈ 𝑋 → 𝑥 ⊆ 𝑧)))
23	12, 13, 22	ralrimd 3218	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧))
24	23	alimi 1808	. . . 4 ⊢ (∀𝑧∀𝑥 ∈ 𝑋 (∀𝑤(𝑤 ⊆ 𝑥 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑥 ⊆ 𝑧) → ∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧))
25	11, 24	syl 17	. . 3 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧))
26		unissb 4862	. . . . 5 ⊢ (∪ 𝑋 ⊆ 𝑧 ↔ ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧)
27	26	imbi2i 338	. . . 4 ⊢ ((∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∪ 𝑋 ⊆ 𝑧) ↔ (∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧))
28	27	albii 1816	. . 3 ⊢ (∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∪ 𝑋 ⊆ 𝑧) ↔ ∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∀𝑥 ∈ 𝑋 𝑥 ⊆ 𝑧))
29	25, 28	sylibr 236	. 2 ⊢ (𝜑 → ∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∪ 𝑋 ⊆ 𝑧))
30		setrec1lem2.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
31	30	uniexd 7462	. . 3 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
32	4, 31	setrec1lem1 44784	. 2 ⊢ (𝜑 → (∪ 𝑋 ∈ 𝑌 ↔ ∀𝑧(∀𝑤(𝑤 ⊆ ∪ 𝑋 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → ∪ 𝑋 ⊆ 𝑧)))
33	29, 32	mpbird 259	1 ⊢ (𝜑 → ∪ 𝑋 ∈ 𝑌)

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  Vcvv 3494   ⊆ wss 3935  ∪ cuni 4831  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-uni 4832

This theorem is referenced by:  setrec1lem3  44786