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Theorem isdir 18447
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 5730 . . . 4 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 unieq 4874 . . . . . . . 8 (𝑟 = 𝑅 𝑟 = 𝑅)
32unieqd 4877 . . . . . . 7 (𝑟 = 𝑅 𝑟 = 𝑅)
4 isdir.1 . . . . . . 7 𝐴 = 𝑅
53, 4eqtr4di 2795 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝐴)
65reseq2d 5935 . . . . 5 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝐴))
7 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
86, 7sseq12d 3975 . . . 4 (𝑟 = 𝑅 → (( I ↾ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
91, 8anbi12d 631 . . 3 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
107, 7coeq12d 5818 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1110, 7sseq12d 3975 . . . 4 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
125sqxpeqd 5663 . . . . 5 (𝑟 = 𝑅 → ( 𝑟 × 𝑟) = (𝐴 × 𝐴))
13 cnveq 5827 . . . . . 6 (𝑟 = 𝑅𝑟 = 𝑅)
1413, 7coeq12d 5818 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1512, 14sseq12d 3975 . . . 4 (𝑟 = 𝑅 → (( 𝑟 × 𝑟) ⊆ (𝑟𝑟) ↔ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))
1611, 15anbi12d 631 . . 3 (𝑟 = 𝑅 → (((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅))))
179, 16anbi12d 631 . 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 18445 . 2 DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
1917, 18elab2g 3630 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 205  wa 396   = wceq 1541  wcel 2106  wss 3908   cuni 4863   I cid 5528   × cxp 5629  ccnv 5630  cres 5633  ccom 5635  Rel wrel 5636  DirRelcdir 18443
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-res 5643  df-dir 18445
This theorem is referenced by:  reldir  18448  dirdm  18449  dirref  18450  dirtr  18451  dirge  18452  tsrdir  18453  filnetlem3  34790
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