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Theorem inintabss 36903
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 590 . . 3 ((𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)) → ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
3 elinintab 36900 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜑}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)))
4 vex 3176 . . . 4 𝑢 ∈ V
5 elinintrab 36902 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥))))
64, 5ax-mp 5 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
72, 3, 63imtr4i 280 . 2 (𝑢 ∈ (𝐴 {𝑥𝜑}) → 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)})
87ssriv 3572 1 (𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cint 4410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-int 4411
This theorem is referenced by: (None)
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