Theorem seinxp 5729

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 5724	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2	1	ancoms 462	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	2	rabbidva 3419	. . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥})
4	3	eleq1d 2846	. . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V))
5	4	ralbiia 3105	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
6		df-se 5599	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
7		df-se 5599	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥} ∈ V)
8	5, 6, 7	3bitr4i 305	1 ⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)

Syntax hints:   ↔ wb 208   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ∩ cin 3903   class class class wbr 5099   Se wse 5596   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-se 5599  df-xp 5651

This theorem is referenced by: (None)