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Theorem sess2 5558
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess2 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))

Proof of Theorem sess2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3987 . . 3 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V))
2 rabss2 4011 . . . . 5 (𝐴𝐵 → {𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥})
3 ssexg 5247 . . . . . 6 (({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} ∧ {𝑦𝐵𝑦𝑅𝑥} ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
43ex 413 . . . . 5 ({𝑦𝐴𝑦𝑅𝑥} ⊆ {𝑦𝐵𝑦𝑅𝑥} → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
52, 4syl 17 . . . 4 (𝐴𝐵 → ({𝑦𝐵𝑦𝑅𝑥} ∈ V → {𝑦𝐴𝑦𝑅𝑥} ∈ V))
65ralimdv 3109 . . 3 (𝐴𝐵 → (∀𝑥𝐴 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
71, 6syld 47 . 2 (𝐴𝐵 → (∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V))
8 df-se 5545 . 2 (𝑅 Se 𝐵 ↔ ∀𝑥𝐵 {𝑦𝐵𝑦𝑅𝑥} ∈ V)
9 df-se 5545 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 93imtr4g 296 1 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3064  {crab 3068  Vcvv 3432  wss 3887   class class class wbr 5074   Se wse 5542
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-se 5545
This theorem is referenced by:  seeq2  5562  wereu2  5586  frpomin  6243  fprlem1  8116  wfrlem5OLD  8144  frmin  9507  frrlem15  9515
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