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.

Step	Hyp	Ref	Expression
1		simpll 481	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → x ∈ A)
2		brinxp 4351	. . . . . . . 8 ⊢ ((x ∈ A ∧ x ∈ A) → (x𝑅x ↔ x(𝑅 ∩ (A × A))x))
3	1, 1, 2	syl2anc 391	. . . . . . 7 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → (x𝑅x ↔ x(𝑅 ∩ (A × A))x))
4	3	notbid 591	. . . . . 6 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → (¬ x𝑅x ↔ ¬ x(𝑅 ∩ (A × A))x))
5		brinxp 4351	. . . . . . . . 9 ⊢ ((x ∈ A ∧ y ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
6	5	adantr 261	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
7		brinxp 4351	. . . . . . . . 9 ⊢ ((y ∈ A ∧ z ∈ A) → (y𝑅z ↔ y(𝑅 ∩ (A × A))z))
8	7	adantll 445	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → (y𝑅z ↔ y(𝑅 ∩ (A × A))z))
9	6, 8	anbi12d 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) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
11	10	adantlr 446	. . . . . . 7 ⊢ (((x ∈ A ∧ y ∈ A) ∧ z ∈ A) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
12	9, 11	imbi12d 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)))
13	4, 12	anbi12d 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))))
14	13	ralbidva 2316	. . . 4 ⊢ ((x ∈ A ∧ y ∈ A) → (∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) ↔ ∀z ∈ A (¬ x(𝑅 ∩ (A × A))x ∧ ((x(𝑅 ∩ (A × A))y ∧ y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z))))
15	14	ralbidva 2316	. . 3 ⊢ (x ∈ A → (∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) ↔ ∀y ∈ A ∀z ∈ A (¬ x(𝑅 ∩ (A × A))x ∧ ((x(𝑅 ∩ (A × A))y ∧ y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z))))
16	15	ralbiia 2332	. 2 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x(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 A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
18		df-po 4024	. 2 ⊢ ((𝑅 ∩ (A × A)) Po A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x(𝑅 ∩ (A × A))x ∧ ((x(𝑅 ∩ (A × A))y ∧ y(𝑅 ∩ (A × A))z) → x(𝑅 ∩ (A × A))z)))
19	16, 17, 18	3bitr4i 201	1 ⊢ (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)

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-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-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