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Theorem sess1 5602
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 484 . . . . . 6 ((𝑅𝑆𝑦𝐴) → 𝑅𝑆)
21ssbrd 5149 . . . . 5 ((𝑅𝑆𝑦𝐴) → (𝑦𝑅𝑥𝑦𝑆𝑥))
32ss2rabdv 4034 . . . 4 (𝑅𝑆 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥})
4 ssexg 5281 . . . . 5 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} ∧ {𝑦𝐴𝑦𝑆𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
54ex 414 . . . 4 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
63, 5syl 17 . . 3 (𝑅𝑆 → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
76ralimdv 3167 . 2 (𝑅𝑆 → (∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5590 . 2 (𝑆 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V)
9 df-se 5590 . 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 397  wcel 2107  wral 3065  {crab 3408  Vcvv 3446  wss 3911   class class class wbr 5106   Se wse 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rab 3409  df-v 3448  df-in 3918  df-ss 3928  df-br 5107  df-se 5590
This theorem is referenced by:  seeq1  5606
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