Theorem seinxp 4334

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 4331	. . . . . 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 2522	. . . 4 ⊢ (x ∈ A → {y ∈ A ∣ y𝑅x} = {y ∈ A ∣ y(𝑅 ∩ (A × A))x})
4	3	eleq1d 2084	. . 3 ⊢ (x ∈ A → ({y ∈ A ∣ y𝑅x} ∈ V ↔ {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V))
5	4	ralbiia 2312	. 2 ⊢ (∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V ↔ ∀x ∈ A {y ∈ A ∣ y(𝑅 ∩ (A × A))x} ∈ V)
6		df-se 4035	. 2 ⊢ (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V)
7		df-se 4035	. 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 1370  ∀wral 2280  {crab 2284  Vcvv 2531   ∩ cin 2889   class class class wbr 3734   Se wse 4034   × cxp 4266

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-se 4035  df-xp 4274

This theorem is referenced by: (None)