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Theorem intabssd 39870
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 2899 . . . . . . . 8 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
43biimpd 231 . . . . . . 7 (𝑥 = 𝐴 → (𝑧𝑥𝑧𝐴))
5 intabssd.ss . . . . . . . 8 (𝜑𝐴𝑦)
65sseld 3964 . . . . . . 7 (𝜑 → (𝑧𝐴𝑧𝑦))
74, 6sylan9r 511 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑧𝑥𝑧𝑦))
82, 7imim12d 81 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
91, 8spcimdv 3590 . . . 4 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → (𝜒𝑧𝑦)))
109alrimdv 1923 . . 3 (𝜑 → (∀𝑥(𝜓𝑧𝑥) → ∀𝑦(𝜒𝑧𝑦)))
11 vex 3496 . . . 4 𝑧 ∈ V
1211elintab 4878 . . 3 (𝑧 {𝑥𝜓} ↔ ∀𝑥(𝜓𝑧𝑥))
1311elintab 4878 . . 3 (𝑧 {𝑦𝜒} ↔ ∀𝑦(𝜒𝑧𝑦))
1410, 12, 133imtr4g 298 . 2 (𝜑 → (𝑧 {𝑥𝜓} → 𝑧 {𝑦𝜒}))
1514ssrdv 3971 1 (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1528   = wceq 1530  wcel 2107  {cab 2797  wss 3934   cint 4867
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-in 3941  df-ss 3950  df-int 4868
This theorem is referenced by:  harval3  39889  clcnvlem  39968
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