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Theorem coeq0 5682
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 5674 and coundir 5675 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 5671 . . 3 Rel (𝐴𝐵)
2 relrn0 5415 . . 3 (Rel (𝐴𝐵) → ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅))
31, 2ax-mp 5 . 2 ((𝐴𝐵) = ∅ ↔ ran (𝐴𝐵) = ∅)
4 rnco 5679 . . 3 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
54eqeq1i 2656 . 2 (ran (𝐴𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6 relres 5461 . . . 4 Rel (𝐴 ↾ ran 𝐵)
7 reldm0 5375 . . . 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 5415 . . . 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 5454 . . . . 5 dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12 incom 3838 . . . . 5 (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
1311, 12eqtri 2673 . . . 4 dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
1413eqeq1i 2656 . . 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 1523  cin 3606  c0 3948  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155
This theorem is referenced by:  coemptyd  13764  coeq0i  37633  diophrw  37639  relexpnul  38287
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