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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.
StepHypRef Expression
1 brinxp 4331 . . . . . 6 ((y A x A) → (y𝑅xy(𝑅 ∩ (A × A))x))
21ancoms 255 . . . . 5 ((x A y A) → (y𝑅xy(𝑅 ∩ (A × A))x))
32rabbidva 2522 . . . 4 (x A → {y Ay𝑅x} = {y Ay(𝑅 ∩ (A × A))x})
43eleq1d 2084 . . 3 (x A → ({y Ay𝑅x} V ↔ {y Ay(𝑅 ∩ (A × A))x} V))
54ralbiia 2312 . 2 (x A {y Ay𝑅x} V ↔ x A {y Ay(𝑅 ∩ (A × A))x} V)
6 df-se 4035 . 2 (𝑅 Se Ax A {y Ay𝑅x} V)
7 df-se 4035 . 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 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)
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