Theorem cllem0 44106

Description: The class of all sets with property 𝜑(𝑧) is closed under the binary operation on sets defined in 𝑅(𝑥, 𝑦). (Contributed by RP, 3-Jan-2020.)

Hypotheses

Ref	Expression
cllem0.v	⊢ 𝑉 = {𝑧 ∣ 𝜑}
cllem0.rex	⊢ 𝑅 ∈ 𝑈
cllem0.r	⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓))
cllem0.x	⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒))
cllem0.y	⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃))
cllem0.closed	⊢ ((𝜒 ∧ 𝜃) → 𝜓)

Assertion

Ref	Expression
cllem0	⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉

Distinct variable groups:   𝜓,𝑧   𝜒,𝑧   𝜃,𝑧   𝑥,𝑦,𝑧   𝑦,𝑉   𝑧,𝑅

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem cllem0

Step	Hyp	Ref	Expression
1		cllem0.rex	. . . . . . 7 ⊢ 𝑅 ∈ 𝑈
2	1	elexi 3475	. . . . . 6 ⊢ 𝑅 ∈ V
3		cllem0.r	. . . . . 6 ⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓))
4		cllem0.v	. . . . . 6 ⊢ 𝑉 = {𝑧 ∣ 𝜑}
5	2, 3, 4	elab2 3641	. . . . 5 ⊢ (𝑅 ∈ 𝑉 ↔ 𝜓)
6	5	ralbii 3107	. . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑉 𝜓)
7	6	ralbii 3107	. . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓)
8		df-ral 3076	. . . 4 ⊢ (∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))
9	8	ralbii 3107	. . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝜓 ↔ ∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))
10		df-ral 3076	. . 3 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦(𝑦 ∈ 𝑉 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)))
11	7, 9, 10	3bitri 299	. 2 ⊢ (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉 ↔ ∀𝑥(𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓)))
12		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
13		cllem0.x	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒))
14	12, 13, 4	elab2 3641	. . . . 5 ⊢ (𝑥 ∈ 𝑉 ↔ 𝜒)
15		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
16		cllem0.y	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃))
17	15, 16, 4	elab2 3641	. . . . 5 ⊢ (𝑦 ∈ 𝑉 ↔ 𝜃)
18		cllem0.closed	. . . . 5 ⊢ ((𝜒 ∧ 𝜃) → 𝜓)
19	14, 17, 18	syl2anb 607	. . . 4 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝜓)
20	19	ex 416	. . 3 ⊢ (𝑥 ∈ 𝑉 → (𝑦 ∈ 𝑉 → 𝜓))
21	20	alrimiv 1946	. 2 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦(𝑦 ∈ 𝑉 → 𝜓))
22	11, 21	mpgbir 1818	1 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-v 3455

This theorem is referenced by:  superficl  44107  superuncl  44108  ssficl  44109  ssuncl  44110  ssdifcl  44111  sssymdifcl  44112  trficl  44209