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Theorem isdir 17055
 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 (𝑅𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))))

Proof of Theorem isdir
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 releq 5124 . . . 4 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 unieq 4380 . . . . . . . 8 (𝑟 = 𝑅 𝑟 = 𝑅)
32unieqd 4382 . . . . . . 7 (𝑟 = 𝑅 𝑟 = 𝑅)
4 isdir.1 . . . . . . 7 𝐴 = 𝑅
53, 4syl6eqr 2662 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝐴)
65reseq2d 5317 . . . . 5 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝐴))
7 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
86, 7sseq12d 3597 . . . 4 (𝑟 = 𝑅 → (( I ↾ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
91, 8anbi12d 743 . . 3 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
107, 7coeq12d 5208 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1110, 7sseq12d 3597 . . . 4 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
125sqxpeqd 5065 . . . . 5 (𝑟 = 𝑅 → ( 𝑟 × 𝑟) = (𝐴 × 𝐴))
13 cnveq 5218 . . . . . 6 (𝑟 = 𝑅𝑟 = 𝑅)
1413, 7coeq12d 5208 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1512, 14sseq12d 3597 . . . 4 (𝑟 = 𝑅 → (( 𝑟 × 𝑟) ⊆ (𝑟𝑟) ↔ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))
1611, 15anbi12d 743 . . 3 (𝑟 = 𝑅 → (((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅))))
179, 16anbi12d 743 . 2 (𝑟 = 𝑅 → (((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟))) ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))))
18 df-dir 17053 . 2 DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
1917, 18elab2g 3322 1 (𝑅𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372   I cid 4948   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  DirRelcdir 17051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-res 5050  df-dir 17053 This theorem is referenced by:  reldir  17056  dirdm  17057  dirref  17058  dirtr  17059  dirge  17060  tsrdir  17061  filnetlem3  31545
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