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Theorem sess1 5650
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess1 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))

Proof of Theorem sess1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . . . 6 ((𝑅𝑆𝑦𝐴) → 𝑅𝑆)
21ssbrd 5186 . . . . 5 ((𝑅𝑆𝑦𝐴) → (𝑦𝑅𝑥𝑦𝑆𝑥))
32ss2rabdv 4076 . . . 4 (𝑅𝑆 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥})
4 ssexg 5323 . . . . 5 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} ∧ {𝑦𝐴𝑦𝑆𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
54ex 412 . . . 4 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
63, 5syl 17 . . 3 (𝑅𝑆 → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
76ralimdv 3169 . 2 (𝑅𝑆 → (∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5638 . 2 (𝑆 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V)
9 df-se 5638 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 296 1 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   class class class wbr 5143   Se wse 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-br 5144  df-se 5638
This theorem is referenced by:  seeq1  5655
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