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Theorem seinxp 5337
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 5333 . . . . . 6 ((𝑦𝐴𝑥𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
21ancoms 447 . . . . 5 ((𝑥𝐴𝑦𝐴) → (𝑦𝑅𝑥𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
32rabbidva 3342 . . . 4 (𝑥𝐴 → {𝑦𝐴𝑦𝑅𝑥} = {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
43eleq1d 2838 . . 3 (𝑥𝐴 → ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
54ralbiia 3131 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6 df-se 5223 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
7 df-se 5223 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
85, 6, 73bitr4i 293 1 (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2148  wral 3064  {crab 3068  Vcvv 3355  cin 3728   class class class wbr 4797   Se wse 5220   × cxp 5261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-br 4798  df-opab 4860  df-se 5223  df-xp 5269
This theorem is referenced by: (None)
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