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Theorem inintabss 42905
Description: Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
inintabss (𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 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.
StepHypRef Expression
1 ax-1 6 . . . 4 (𝑢𝐴 → (∃𝑥𝜑𝑢𝐴))
21anim1i 614 . . 3 ((𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)) → ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
3 elinintab 42902 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜑}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)))
4 elinintrab 42904 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥))))
54elv 3474 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
62, 3, 53imtr4i 292 . 2 (𝑢 ∈ (𝐴 {𝑥𝜑}) → 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)})
76ssriv 3981 1 (𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2703  {crab 3426  Vcvv 3468  cin 3942  wss 3943  𝒫 cpw 4597   cint 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-int 4944
This theorem is referenced by: (None)
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