Theorem poinxp 5658

Description: Intersection of partial order with Cartesian 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		brinxp 5656	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2	1	anidms 566	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	2	ad2antrr 722	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
4	3	notbid 317	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5		brinxp 5656	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6	5	adantr 480	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7		brinxp 5656	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
8	7	adantll 710	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
9	6, 8	anbi12d 630	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10		brinxp 5656	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
11	10	adantlr 711	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
12	9, 11	imbi12d 344	. . . . . 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 630	. . . . 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 3119	. . . 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 3119	. . 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 3089	. 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 5494	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18		df-po 5494	. 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 302	1 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∀wral 3063   ∩ cin 3882   class class class wbr 5070   Po wpo 5492   × cxp 5578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-po 5494  df-xp 5586

This theorem is referenced by:  soinxp  5659