Theorem seinxp 4354

Description: Intersection of set-like relation with cross 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 4351	. . . . . 6 ⊢ ((y ∈ A ∧ x ∈ A) → (y𝑅x ↔ y(𝑅 ∩ (A × A))x))
2	1	ancoms 255	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ A) → (y𝑅x ↔ y(𝑅 ∩ (A × A))x))
3	2	rabbidva 2542	. . . 4 ⊢ (x ∈ A → {y ∈ A ∣ y𝑅x} = {y ∈ A ∣ y(𝑅 ∩ (A × A))x})
4	3	eleq1d 2103	. . 3 ⊢ (x ∈ A → ({y ∈ A ∣ y𝑅x} ∈ V ↔ {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V))
5	4	ralbiia 2332	. 2 ⊢ (∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V ↔ ∀x ∈ A {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V)
6		df-se 4056	. 2 ⊢ (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V)
7		df-se 4056	. 2 ⊢ ((𝑅 ∩ (A × A)) Se A ↔ ∀x ∈ A {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V)
8	5, 6, 7	3bitr4i 201	1 ⊢ (𝑅 Se A ↔ (𝑅 ∩ (A × A)) Se A)

Syntax hints:   ↔ wb 98   ∈ wcel 1390  ∀wral 2300  {crab 2304  Vcvv 2551   ∩ cin 2910   class class class wbr 3755   Se wse 4055   × cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-se 4056  df-xp 4294

This theorem is referenced by: (None)