Theorem poinxp 4697

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 527	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ 𝐴)
2		brinxp 4696	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	1, 1, 2	syl2anc 411	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
4	3	notbid 667	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5		brinxp 4696	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6	5	adantr 276	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7		brinxp 4696	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
8	7	adantll 476	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
9	6, 8	anbi12d 473	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10		brinxp 4696	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
11	10	adantlr 477	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
12	9, 11	imbi12d 234	. . . . . 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 473	. . . . 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 2473	. . . 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 2473	. . 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 2491	. 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 4298	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18		df-po 4298	. 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 212	1 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  ∀wral 2455   ∩ cin 3130   class class class wbr 4005   Po wpo 4296   × cxp 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-po 4298  df-xp 4634

This theorem is referenced by:  soinxp  4698