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Theorem poinxp 4352
Description: Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
poinxp (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)

Proof of Theorem poinxp
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 481 . . . . . . . 8 (((x A y A) z A) → x A)
2 brinxp 4351 . . . . . . . 8 ((x A x A) → (x𝑅xx(𝑅 ∩ (A × A))x))
31, 1, 2syl2anc 391 . . . . . . 7 (((x A y A) z A) → (x𝑅xx(𝑅 ∩ (A × A))x))
43notbid 591 . . . . . 6 (((x A y A) z A) → (¬ x𝑅x ↔ ¬ x(𝑅 ∩ (A × A))x))
5 brinxp 4351 . . . . . . . . 9 ((x A y A) → (x𝑅yx(𝑅 ∩ (A × A))y))
65adantr 261 . . . . . . . 8 (((x A y A) z A) → (x𝑅yx(𝑅 ∩ (A × A))y))
7 brinxp 4351 . . . . . . . . 9 ((y A z A) → (y𝑅zy(𝑅 ∩ (A × A))z))
87adantll 445 . . . . . . . 8 (((x A y A) z A) → (y𝑅zy(𝑅 ∩ (A × A))z))
96, 8anbi12d 442 . . . . . . 7 (((x A y A) z A) → ((x𝑅y y𝑅z) ↔ (x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z)))
10 brinxp 4351 . . . . . . . 8 ((x A z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
1110adantlr 446 . . . . . . 7 (((x A y A) z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
129, 11imbi12d 223 . . . . . 6 (((x A y A) z A) → (((x𝑅y y𝑅z) → x𝑅z) ↔ ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z)))
134, 12anbi12d 442 . . . . 5 (((x A y A) z A) → ((¬ x𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ (¬ x(𝑅 ∩ (A × A))x ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z))))
1413ralbidva 2316 . . . 4 ((x A y A) → (z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ z Ax(𝑅 ∩ (A × A))x ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z))))
1514ralbidva 2316 . . 3 (x A → (y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ y A z Ax(𝑅 ∩ (A × A))x ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z))))
1615ralbiia 2332 . 2 (x A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)) ↔ x A y A z Ax(𝑅 ∩ (A × A))x ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z)))
17 df-po 4024 . 2 (𝑅 Po Ax A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
18 df-po 4024 . 2 ((𝑅 ∩ (A × A)) Po Ax A y A z Ax(𝑅 ∩ (A × A))x ((x(𝑅 ∩ (A × A))y y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z)))
1916, 17, 183bitr4i 201 1 (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wcel 1390  wral 2300  cin 2910   class class class wbr 3755   Po wpo 4022   × 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-in1 544  ax-in2 545  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-bnd 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-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-po 4024  df-xp 4294
This theorem is referenced by:  soinxp  4353
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