Theorem coeq0 6148

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 6140 and coundir 6141 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
coeq0	⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Proof of Theorem coeq0

Step	Hyp	Ref	Expression
1		relco 6137	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relrn0 5867	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)
4		rnco 6145	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
5	4	eqeq1i 2743	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6		relres 5909	. . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵)
7		reldm0 5826	. . . 4 ⊢ (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅))
8	6, 7	ax-mp 5	. . 3 ⊢ ((𝐴 ↾ ran 𝐵) = ∅ ↔ dom (𝐴 ↾ ran 𝐵) = ∅)
9		relrn0 5867	. . . 4 ⊢ (Rel (𝐴 ↾ ran 𝐵) → ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅))
10	6, 9	ax-mp 5	. . 3 ⊢ ((𝐴 ↾ ran 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
11		dmres 5902	. . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12		incom 4131	. . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
13	11, 12	eqtri 2766	. . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
14	13	eqeq1i 2743	. . 3 ⊢ (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
15	8, 10, 14	3bitr3i 300	. 2 ⊢ (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
16	3, 5, 15	3bitri 296	1 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Syntax hints:   ↔ wb 205   = wceq 1539   ∩ cin 3882  ∅c0 4253  dom cdm 5580  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592

This theorem is referenced by:  coemptyd  14618  diophrw  40497  relexpnul  41175