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Theorem isdir 17512
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 5373 . . . 4 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 unieq 4604 . . . . . . . 8 (𝑟 = 𝑅 𝑟 = 𝑅)
32unieqd 4606 . . . . . . 7 (𝑟 = 𝑅 𝑟 = 𝑅)
4 isdir.1 . . . . . . 7 𝐴 = 𝑅
53, 4syl6eqr 2817 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝐴)
65reseq2d 5567 . . . . 5 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝐴))
7 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
86, 7sseq12d 3796 . . . 4 (𝑟 = 𝑅 → (( I ↾ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
91, 8anbi12d 624 . . 3 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
107, 7coeq12d 5457 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1110, 7sseq12d 3796 . . . 4 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
125sqxpeqd 5311 . . . . 5 (𝑟 = 𝑅 → ( 𝑟 × 𝑟) = (𝐴 × 𝐴))
13 cnveq 5466 . . . . . 6 (𝑟 = 𝑅𝑟 = 𝑅)
1413, 7coeq12d 5457 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1512, 14sseq12d 3796 . . . 4 (𝑟 = 𝑅 → (( 𝑟 × 𝑟) ⊆ (𝑟𝑟) ↔ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))
1611, 15anbi12d 624 . . 3 (𝑟 = 𝑅 → (((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅))))
179, 16anbi12d 624 . 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 17510 . 2 DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
1917, 18elab2g 3510 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 197  wa 384   = wceq 1652  wcel 2155  wss 3734   cuni 4596   I cid 5186   × cxp 5277  ccnv 5278  cres 5281  ccom 5283  Rel wrel 5284  DirRelcdir 17508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-v 3352  df-in 3741  df-ss 3748  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-res 5291  df-dir 17510
This theorem is referenced by:  reldir  17513  dirdm  17514  dirref  17515  dirtr  17516  dirge  17517  tsrdir  17518  filnetlem3  32839
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