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Theorem isdir 17001
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 5114 . . . 4 (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅))
2 unieq 4374 . . . . . . . 8 (𝑟 = 𝑅 𝑟 = 𝑅)
32unieqd 4376 . . . . . . 7 (𝑟 = 𝑅 𝑟 = 𝑅)
4 isdir.1 . . . . . . 7 𝐴 = 𝑅
53, 4syl6eqr 2661 . . . . . 6 (𝑟 = 𝑅 𝑟 = 𝐴)
65reseq2d 5304 . . . . 5 (𝑟 = 𝑅 → ( I ↾ 𝑟) = ( I ↾ 𝐴))
7 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
86, 7sseq12d 3596 . . . 4 (𝑟 = 𝑅 → (( I ↾ 𝑟) ⊆ 𝑟 ↔ ( I ↾ 𝐴) ⊆ 𝑅))
91, 8anbi12d 742 . . 3 (𝑟 = 𝑅 → ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ↔ (Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅)))
107, 7coeq12d 5196 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1110, 7sseq12d 3596 . . . 4 (𝑟 = 𝑅 → ((𝑟𝑟) ⊆ 𝑟 ↔ (𝑅𝑅) ⊆ 𝑅))
125sqxpeqd 5055 . . . . 5 (𝑟 = 𝑅 → ( 𝑟 × 𝑟) = (𝐴 × 𝐴))
13 cnveq 5206 . . . . . 6 (𝑟 = 𝑅𝑟 = 𝑅)
1413, 7coeq12d 5196 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟) = (𝑅𝑅))
1512, 14sseq12d 3596 . . . 4 (𝑟 = 𝑅 → (( 𝑟 × 𝑟) ⊆ (𝑟𝑟) ↔ (𝐴 × 𝐴) ⊆ (𝑅𝑅)))
1611, 15anbi12d 742 . . 3 (𝑟 = 𝑅 → (((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)) ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (𝑅𝑅))))
179, 16anbi12d 742 . 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 16999 . 2 DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ 𝑟) ⊆ 𝑟) ∧ ((𝑟𝑟) ⊆ 𝑟 ∧ ( 𝑟 × 𝑟) ⊆ (𝑟𝑟)))}
1917, 18elab2g 3321 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 194  wa 382   = wceq 1474  wcel 1976  wss 3539   cuni 4366   I cid 4938   × cxp 5026  ccnv 5027  cres 5030  ccom 5032  Rel wrel 5033  DirRelcdir 16997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367  df-br 4578  df-opab 4638  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-res 5040  df-dir 16999
This theorem is referenced by:  reldir  17002  dirdm  17003  dirref  17004  dirtr  17005  dirge  17006  tsrdir  17007  filnetlem3  31351
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