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Theorem sess1 5654
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 5191 . . . . 5 ((𝑅𝑆𝑦𝐴) → (𝑦𝑅𝑥𝑦𝑆𝑥))
32ss2rabdv 4086 . . . 4 (𝑅𝑆 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐴𝑦𝑆𝑥})
4 ssexg 5329 . . . . 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 3167 . 2 (𝑅𝑆 → (∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5642 . 2 (𝑆 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑆𝑥} ∈ V)
9 df-se 5642 . 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 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963   class class class wbr 5148   Se wse 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-br 5149  df-se 5642
This theorem is referenced by:  seeq1  5659
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