Theorem clublem 42816

Description: If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.)

Hypotheses

Ref	Expression
clublem.y	⊢ (𝜑 → 𝑌 ∈ V)
clublem.sub	⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒))
clublem.sup	⊢ (𝜑 → 𝑋 ⊆ 𝑌)
clublem.maj	⊢ (𝜑 → 𝜒)

Assertion

Ref	Expression
clublem	⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)

Distinct variable groups:   𝜒,𝑥   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem clublem

Step	Hyp	Ref	Expression
1		clublem.sup	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑌)
2		clublem.maj	. . 3 ⊢ (𝜑 → 𝜒)
3		clublem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ V)
4	3	a1d 25	. . . 4 ⊢ (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝜒) → 𝑌 ∈ V))
5		clublem.sub	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒))
6	5	cleq2lem 42814	. . . . 5 ⊢ (𝑥 = 𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒)))
7	6	elab3g 3667	. . . 4 ⊢ (((𝑋 ⊆ 𝑌 ∧ 𝜒) → 𝑌 ∈ V) → (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒)))
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒)))
9	1, 2, 8	mpbir2and 710	. 2 ⊢ (𝜑 → 𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
10		intss1 4957	. 2 ⊢ (𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)
11	9, 10	syl 17	1 ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2701  Vcvv 3466   ⊆ wss 3940  ∩ cint 4940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957  df-int 4941

This theorem is referenced by:  mptrcllem  42819  trclubgNEW  42824