Theorem clublem 43600

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 43598	. . . . 5 ⊢ (𝑥 = 𝑌 → ((𝑋 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑋 ⊆ 𝑌 ∧ 𝜒)))
7	6	elab3g 3668	. . . 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 713	. 2 ⊢ (𝜑 → 𝑌 ∈ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
10		intss1 4943	. 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 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2712  Vcvv 3463   ⊆ wss 3931  ∩ cint 4926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-ss 3948  df-int 4927

This theorem is referenced by:  mptrcllem  43603  trclubgNEW  43608