Theorem inintabss 44840

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 627	. . 3 ⊢ ((𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)) → ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
3		elinintab 44837	. . 3 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝑢 ∈ 𝐴 ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
4		elinintrab 44839	. . . 4 ⊢ (𝑢 ∈ V → (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥))))
5	4	elv 3459	. . 3 ⊢ (𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝑢 ∈ 𝐴) ∧ ∀𝑥(𝜑 → 𝑢 ∈ 𝑥)))
6	2, 3, 5	3imtr4i 295	. 2 ⊢ (𝑢 ∈ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) → 𝑢 ∈ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)})
7	6	ssriv 3917	1 ⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∩ cint 4912

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562  df-int 4913

This theorem is used by: (None)