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Theorem seinxp 5769
Description: Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
seinxp (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Proof of Theorem seinxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brinxp 5764 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21ancoms 458 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32rabbidva 3443 . . . 4 (𝑥𝐴 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
43eleq1d 2826 . . 3 (𝑥𝐴 → ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
54ralbiia 3091 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6 df-se 5638 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
7 df-se 5638 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
85, 6, 73bitr4i 303 1 (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cin 3950   class class class wbr 5143   Se wse 5635   × cxp 5683
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-se 5638  df-xp 5691
This theorem is referenced by: (None)
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