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Theorem sess1 4100
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 107 . . . . . 6 ((𝑅𝑆𝑦𝐴) → 𝑅𝑆)
21ssbrd 3834 . . . . 5 ((𝑅𝑆𝑦𝐴) → (𝑦𝑅𝑥𝑦𝑆𝑥))
32ss2rabdv 3076 . . . 4 (𝑅𝑆 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥})
4 ssexg 3925 . . . . 5 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} ∧ {𝑦𝐴𝑦𝑆𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
54ex 113 . . . 4 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥} → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
63, 5syl 14 . . 3 (𝑅𝑆 → ({𝑦𝐴𝑦𝑆𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
76ralimdv 2431 . 2 (𝑅𝑆 → (∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 4096 . 2 (𝑆 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V)
9 df-se 4096 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 203 1 (𝑅𝑆 → (𝑆 Se 𝐴𝑅 Se 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  wral 2349  {crab 2353  Vcvv 2602  wss 2974   class class class wbr 3793   Se wse 4092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-br 3794  df-se 4096
This theorem is referenced by:  seeq1  4102
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