Theorem inintabss 43591

Description: Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
inintabss	⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}

Distinct variable groups:   𝜑,𝑤   𝑥,𝑤,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem inintabss

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 ⊢ (𝑢 ∈ 𝐴 → (∃𝑥𝜑 → 𝑢 ∈ 𝐴))
2	1	anim1i 615	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)) → ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
3		elinintab 43588	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
4		elinintrab 43590	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))))
5	4	elv 3485	. . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
6	2, 3, 5	3imtr4i 292	. 2 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) → 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)})
7	6	ssriv 3987	1 ⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∩ cint 4946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-int 4947

This theorem is referenced by: (None)