Theorem isdir 18533

Description: A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
isdir.1	⊢ 𝐴 = ∪ ∪ 𝑅

Assertion

Ref	Expression
isdir	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))))

Proof of Theorem isdir

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		releq 5734	. . . 4 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2		unieq 4876	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅)
3	2	unieqd 4878	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅)
4		isdir.1	. . . . . . 7 ⊢ 𝐴 = ∪ ∪ 𝑅
5	3, 4	eqtr4di 2790	. . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = 𝐴)
6	5	reseq2d 5946	. . . . 5 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ 𝐴))
7		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
8	6, 7	sseq12d 3969	. . . 4 ⊢ (𝑟 = 𝑅 → (( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
9	1, 8	anbi12d 633	. . 3 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
10	7, 7	coeq12d 5821	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅))
11	10, 7	sseq12d 3969	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅))
12	5	sqxpeqd 5664	. . . . 5 ⊢ (𝑟 = 𝑅 → (∪ ∪ 𝑟 × ∪ ∪ 𝑟) = (𝐴 × 𝐴))
13		cnveq 5830	. . . . . 6 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅)
14	13, 7	coeq12d 5821	. . . . 5 ⊢ (𝑟 = 𝑅 → (◡𝑟 ∘ 𝑟) = (◡𝑅 ∘ 𝑅))
15	12, 14	sseq12d 3969	. . . 4 ⊢ (𝑟 = 𝑅 → ((∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟) ↔ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))
16	11, 15	anbi12d 633	. . 3 ⊢ (𝑟 = 𝑅 → (((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)) ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅))))
17	9, 16	anbi12d 633	. 2 ⊢ (𝑟 = 𝑅 → (((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟))) ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))))
18		df-dir 18531	. 2 ⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))}
19	17, 18	elab2g 3637	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ∪ cuni 4865   I cid 5526   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   ∘ ccom 5636  Rel wrel 5637  DirRelcdir 18529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-res 5644  df-dir 18531

This theorem is referenced by:  reldir  18534  dirdm  18535  dirref  18536  dirtr  18537  dirge  18538  tsrdir  18539  filnetlem3  36596