Theorem seinxp 5661

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 5656	. . . . . 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 3402	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4	3	eleq1d 2823	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
5	4	ralbiia 3089	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6		df-se 5536	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
7		df-se 5536	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
8	5, 6, 7	3bitr4i 302	1 ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Syntax hints:   ↔ wb 205   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   ∩ cin 3882   class class class wbr 5070   Se wse 5533   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-se 5536  df-xp 5586

This theorem is referenced by: (None)