Theorem coeq0 6259

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 6250 and coundir 6251 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 6112	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relrn0 5965	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)
4		rnco 6255	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
5	4	eqeq1i 2770	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6		relres 6006	. . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵)
7		reldm0 5920	. . . 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 5965	. . . 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 6013	. . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12		incom 4162	. . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
13	11, 12	eqtri 2788	. . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
14	13	eqeq1i 2770	. . 3 ⊢ (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
15	8, 10, 14	3bitr3i 304	. 2 ⊢ (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
16	3, 5, 15	3bitri 300	1 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∩ cin 3905  ∅c0 4286  dom cdm 5663  ran crn 5664   ↾ cres 5665   ∘ ccom 5667  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by:  coemptyd  15036  wrdpmtrlast  33453  diophrw  43523  relexpnul  44437