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Theorem coeq0 6275
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 6267 and coundir 6268 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 6126 . . 3 Rel (𝐴𝐵)
2 relrn0 5983 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 rnco 6272 . . 3 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
54eqeq1i 2742 . 2 (ran (𝐴𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6 relres 6023 . . . 4 Rel (𝐴 ↾ ran 𝐵)
7 reldm0 5938 . . . 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 5983 . . . 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 6030 . . . . 5 dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12 incom 4209 . . . . 5 (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
1311, 12eqtri 2765 . . . 4 dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
1413eqeq1i 2742 . . 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 206   = wceq 1540  cin 3950  c0 4333  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  coemptyd  15018  wrdpmtrlast  33113  diophrw  42770  relexpnul  43691
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