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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.
StepHypRef Expression
1 brinxp 4351 . . . . . 6 ((y A x A) → (y𝑅xy(𝑅 ∩ (A × A))x))
21ancoms 255 . . . . 5 ((x A y A) → (y𝑅xy(𝑅 ∩ (A × A))x))
32rabbidva 2542 . . . 4 (x A → {y Ay𝑅x} = {y Ay(𝑅 ∩ (A × A))x})
43eleq1d 2103 . . 3 (x A → ({y Ay𝑅x} V ↔ {y Ay(𝑅 ∩ (A × A))x} V))
54ralbiia 2332 . 2 (x A {y Ay𝑅x} V ↔ x A {y Ay(𝑅 ∩ (A × A))x} V)
6 df-se 4056 . 2 (𝑅 Se Ax A {y Ay𝑅x} V)
7 df-se 4056 . 2 ((𝑅 ∩ (A × A)) Se Ax A {y Ay(𝑅 ∩ (A × A))x} V)
85, 6, 73bitr4i 201 1 (𝑅 Se A ↔ (𝑅 ∩ (A × A)) Se A)
Colors of variables: wff set class
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)
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