Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  coeq0 Structured version   Visualization version   GIF version

Theorem coeq0 5561
 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 5553 and coundir 5554 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 5550 . . 3 Rel (𝐴𝐵)
2 relrn0 5304 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 rnco 5558 . . 3 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
54eqeq1i 2615 . 2 (ran (𝐴𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6 relres 5346 . . . 4 Rel (𝐴 ↾ ran 𝐵)
7 reldm0 5264 . . . 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 5304 . . . 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 5339 . . . . 5 dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12 incom 3767 . . . . 5 (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
1311, 12eqtri 2632 . . . 4 dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
1413eqeq1i 2615 . . 3 (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
158, 10, 143bitr3i 289 . 2 (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
163, 5, 153bitri 285 1 ((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∩ cin 3539  ∅c0 3874  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by:  coemptyd  13566  coeq0i  36334  diophrw  36340  relexpnul  36989
 Copyright terms: Public domain W3C validator