Theorem inintabss 43617

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 43614	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
4		elinintrab 43616	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))))
5	4	elv 3441	. . 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 3938	1 ⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  {crab 3395  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4550  ∩ cint 4897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-in 3909  df-ss 3919  df-pw 4552  df-int 4898

This theorem is referenced by: (None)