Theorem coeq0 6257

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 6248 and coundir 6249 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 6103	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relrn0 5954	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)
4		rnco 6253	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
5	4	eqeq1i 2765	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6		relres 5995	. . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵)
7		reldm0 5909	. . . 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 5954	. . . 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 6002	. . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12		incom 4106	. . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
13	11, 12	eqtri 2783	. . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
14	13	eqeq1i 2765	. . 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 1569   ∩ cin 4089  ∅c0 4278  dom cdm 5650  ran crn 5651   ↾ cres 5652   ∘ ccom 5654  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662

This theorem is used by:  coemptyd  15138  wrdpmtrlast  33701  diophrw  43864  relexpnul  44772