Theorem poinxp 5728

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 5726	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
2	1	anidms 574	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
3	2	ad2antrr 736	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑥 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
4	3	notbid 320	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5		brinxp 5726	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
6	5	adantr 484	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
7		brinxp 5726	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
8	7	adantll 724	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑦𝑅𝑧 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧))
9	6, 8	anbi12d 641	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∧ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑧)))
10		brinxp 5726	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
11	10	adantlr 725	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑧))
12	9, 11	imbi12d 346	. . . . . 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 641	. . . . 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 3183	. . . 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 3183	. . 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 3106	. 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 5555	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18		df-po 5555	. 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 305	1 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∀wral 3076   ∩ cin 3903   class class class wbr 5100   Po wpo 5553   × cxp 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5555  df-xp 5653

This theorem is referenced by:  soinxp  5729