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Theorem inintabss 37404
 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 591 . . 3 ((𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)) → ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
3 elinintab 37401 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜑}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)))
4 vex 3193 . . . 4 𝑢 ∈ V
5 elinintrab 37403 . . . 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 281 . 2 (𝑢 ∈ (𝐴 {𝑥𝜑}) → 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)})
87ssriv 3592 1 (𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  {crab 2912  Vcvv 3190   ∩ cin 3559   ⊆ wss 3560  𝒫 cpw 4136  ∩ cint 4447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-in 3567  df-ss 3574  df-pw 4138  df-int 4448 This theorem is referenced by: (None)
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