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Theorem inintabss 40069
 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 616 . . 3 ((𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)) → ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
3 elinintab 40066 . . 3 (𝑢 ∈ (𝐴 {𝑥𝜑}) ↔ (𝑢𝐴 ∧ ∀𝑥(𝜑𝑢𝑥)))
4 elinintrab 40068 . . . 4 (𝑢 ∈ V → (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥))))
54elv 3475 . . 3 (𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝑢𝐴) ∧ ∀𝑥(𝜑𝑢𝑥)))
62, 3, 53imtr4i 294 . 2 (𝑢 ∈ (𝐴 {𝑥𝜑}) → 𝑢 {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)})
76ssriv 3946 1 (𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2798  {crab 3129  Vcvv 3470   ∩ cin 3908   ⊆ wss 3909  𝒫 cpw 4511  ∩ cint 4848 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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3472  df-in 3916  df-ss 3926  df-pw 4513  df-int 4849 This theorem is referenced by: (None)
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