Theorem seinxp 5743

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.

Step	Hyp	Ref	Expression
1		brinxp 5738	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2	1	ancoms 458	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	2	rabbidva 3427	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4	3	eleq1d 2820	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
5	4	ralbiia 3081	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6		df-se 5612	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
7		df-se 5612	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
8	5, 6, 7	3bitr4i 303	1 ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2109  ∀wral 3052  {crab 3420  Vcvv 3464   ∩ cin 3930   class class class wbr 5124   Se wse 5609   × cxp 5657

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-se 5612  df-xp 5665

This theorem is referenced by: (None)