Theorem intabssd 40227

Description: When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)

Hypotheses

Ref	Expression
intabssd.ex	⊢ (𝜑 → 𝐴 ∈ 𝑉)
intabssd.sub	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
intabssd.ss	⊢ (𝜑 → 𝐴 ⊆ 𝑦)

Assertion

Ref	Expression
intabssd	⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒})

Distinct variable groups:   𝜒,𝑥   𝜓,𝑦   𝑥,𝑦,𝜑   𝑥,𝐴

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem intabssd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intabssd.ex	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		intabssd.sub	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
3		eleq2 2878	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝐴))
4	3	biimpd 232	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝐴))
5		intabssd.ss	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑦)
6	5	sseld 3914	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ 𝐴 → 𝑧 ∈ 𝑦))
7	4, 6	sylan9r 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦))
8	2, 7	imim12d 81	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝜓 → 𝑧 ∈ 𝑥) → (𝜒 → 𝑧 ∈ 𝑦)))
9	1, 8	spcimdv 3540	. . . 4 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑧 ∈ 𝑥) → (𝜒 → 𝑧 ∈ 𝑦)))
10	9	alrimdv 1930	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑧 ∈ 𝑥) → ∀𝑦(𝜒 → 𝑧 ∈ 𝑦)))
11		vex 3444	. . . 4 ⊢ 𝑧 ∈ V
12	11	elintab 4849	. . 3 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜓 → 𝑧 ∈ 𝑥))
13	11	elintab 4849	. . 3 ⊢ (𝑧 ∈ ∩ {𝑦 ∣ 𝜒} ↔ ∀𝑦(𝜒 → 𝑧 ∈ 𝑦))
14	10, 12, 13	3imtr4g 299	. 2 ⊢ (𝜑 → (𝑧 ∈ ∩ {𝑥 ∣ 𝜓} → 𝑧 ∈ ∩ {𝑦 ∣ 𝜒}))
15	14	ssrdv 3921	1 ⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒})

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776   ⊆ wss 3881  ∩ cint 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-int 4839

This theorem is referenced by:  harval3  40244  clcnvlem  40323