MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seinxp Structured version   Visualization version   GIF version

Theorem seinxp 5761
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 5756 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21ancoms 458 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32rabbidva 3436 . . . 4 (𝑥𝐴 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
43eleq1d 2814 . . 3 (𝑥𝐴 → ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
54ralbiia 3088 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6 df-se 5634 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
7 df-se 5634 . 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 205  wcel 2099  wral 3058  {crab 3429  Vcvv 3471  cin 3946   class class class wbr 5148   Se wse 5631   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-se 5634  df-xp 5684
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator