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Theorem coeq0 6245
Description: A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 6237 and coundir 6238 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
Assertion
Ref Expression
coeq0 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Proof of Theorem coeq0
StepHypRef Expression
1 relco 6098 . . 3 Rel (𝐴𝐵)
2 relrn0 5959 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 rnco 6242 . . 3 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
54eqeq1i 2729 . 2 (ran (𝐴𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6 relres 6001 . . . 4 Rel (𝐴 ↾ ran 𝐵)
7 reldm0 5918 . . . 4 (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅))
86, 7ax-mp 5 . . 3 ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅)
9 relrn0 5959 . . . 4 (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅))
106, 9ax-mp 5 . . 3 ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
11 dmres 5994 . . . . 5 dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12 incom 4194 . . . . 5 (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
1311, 12eqtri 2752 . . . 4 dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
1413eqeq1i 2729 . . 3 (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
158, 10, 143bitr3i 301 . 2 (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
163, 5, 153bitri 297 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  cin 3940  c0 4315  dom cdm 5667  ran crn 5668  cres 5669  ccom 5671  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679
This theorem is referenced by:  coemptyd  14928  diophrw  42047  relexpnul  42978
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