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Theorem coeq0 5603
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 5595 and coundir 5596 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 5592 . . 3 Rel (𝐴𝐵)
2 relrn0 5343 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 rnco 5600 . . 3 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
54eqeq1i 2626 . 2 (ran (𝐴𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6 relres 5385 . . . 4 Rel (𝐴 ↾ ran 𝐵)
7 reldm0 5303 . . . 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 5343 . . . 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 5378 . . . . 5 dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12 incom 3783 . . . . 5 (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
1311, 12eqtri 2643 . . . 4 dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
1413eqeq1i 2626 . . 3 (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
158, 10, 143bitr3i 290 . 2 (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
163, 5, 153bitri 286 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cin 3554  c0 3891  dom cdm 5074  ran crn 5075  cres 5076  ccom 5078  Rel wrel 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086
This theorem is referenced by:  coemptyd  13652  coeq0i  36793  diophrw  36799  relexpnul  37448
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