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Theorem isdir 17433
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 5358 . . . 4 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 unieq 4596 . . . . . . . 8 (𝑟 = 𝑅 𝑟 = 𝑅)
32unieqd 4598 . . . . . . 7 (𝑟 = 𝑅 𝑟 = 𝑅)
4 isdir.1 . . . . . . 7 𝐴 = 𝑅
53, 4syl6eqr 2812 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝐴)
65reseq2d 5551 . . . . 5 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝐴))
7 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
86, 7sseq12d 3775 . . . 4 (𝑟 = 𝑅 → (( I ↾ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
91, 8anbi12d 749 . . 3 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
107, 7coeq12d 5442 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1110, 7sseq12d 3775 . . . 4 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
125sqxpeqd 5298 . . . . 5 (𝑟 = 𝑅 → ( 𝑟 × 𝑟) = (𝐴 × 𝐴))
13 cnveq 5451 . . . . . 6 (𝑟 = 𝑅𝑟 = 𝑅)
1413, 7coeq12d 5442 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1512, 14sseq12d 3775 . . . 4 (𝑟 = 𝑅 → (( 𝑟 × 𝑟) ⊆ (𝑟𝑟) ↔ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))
1611, 15anbi12d 749 . . 3 (𝑟 = 𝑅 → (((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅))))
179, 16anbi12d 749 . 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 17431 . 2 DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
1917, 18elab2g 3493 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 196  wa 383   = wceq 1632  wcel 2139  wss 3715   cuni 4588   I cid 5173   × cxp 5264  ccnv 5265  cres 5268  ccom 5270  Rel wrel 5271  DirRelcdir 17429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-res 5278  df-dir 17431
This theorem is referenced by:  reldir  17434  dirdm  17435  dirref  17436  dirtr  17437  dirge  17438  tsrdir  17439  filnetlem3  32681
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