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Theorem intabssd 38699
Description: When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)
Hypotheses
Ref Expression
intabssd.ex (𝜑𝐴𝑉)
intabssd.sub ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
intabssd.ss (𝜑𝐴𝑦)
Assertion
Ref Expression
intabssd (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
Distinct variable groups:   𝜒,𝑥   𝜓,𝑦   𝑥,𝑦,𝜑   𝑥,𝐴
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem intabssd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 intabssd.ex . . . . 5 (𝜑𝐴𝑉)
2 intabssd.sub . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
3 eleq2 2867 . . . . . . . 8 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
43biimpd 221 . . . . . . 7 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
5 intabssd.ss . . . . . . . 8 (𝜑𝐴𝑦)
65sseld 3797 . . . . . . 7 (𝜑 → (𝑧𝐴𝑧𝑦))
74, 6sylan9r 505 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑧𝑥𝑧𝑦))
82, 7imim12d 81 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
91, 8spcimdv 3478 . . . 4 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
109alrimdv 2025 . . 3 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → ∀𝑦(𝜒𝑧𝑦)))
11 vex 3388 . . . 4 𝑧 ∈ V
1211elintab 4678 . . 3 (𝑧 {𝑥𝜓} ↔ ∀𝑥(𝜓𝑧𝑥))
1311elintab 4678 . . 3 (𝑧 {𝑦𝜒} ↔ ∀𝑦(𝜒𝑧𝑦))
1410, 12, 133imtr4g 288 . 2 (𝜑 → (𝑧 {𝑥𝜓} → 𝑧 {𝑦𝜒}))
1514ssrdv 3804 1 (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651   = wceq 1653  wcel 2157  {cab 2785  wss 3769   cint 4667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-int 4668
This theorem is referenced by:  clcnvlem  38713
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