Theorem coeq0 6222

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 6213 and coundir 6214 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 6075	. . 3 ⊢ Rel (𝐴 ∘ 𝐵)
2		relrn0 5930	. . 3 ⊢ (Rel (𝐴 ∘ 𝐵) → ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ∘ 𝐵) = ∅)
4		rnco 6218	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)
5	4	eqeq1i 2742	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ∅ ↔ ran (𝐴 ↾ ran 𝐵) = ∅)
6		relres 5972	. . . 4 ⊢ Rel (𝐴 ↾ ran 𝐵)
7		reldm0 5885	. . . 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 5930	. . . 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 5979	. . . . 5 ⊢ dom (𝐴 ↾ ran 𝐵) = (ran 𝐵 ∩ dom 𝐴)
12		incom 4163	. . . . 5 ⊢ (ran 𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ ran 𝐵)
13	11, 12	eqtri 2760	. . . 4 ⊢ dom (𝐴 ↾ ran 𝐵) = (dom 𝐴 ∩ ran 𝐵)
14	13	eqeq1i 2742	. . 3 ⊢ (dom (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
15	8, 10, 14	3bitr3i 301	. 2 ⊢ (ran (𝐴 ↾ ran 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
16	3, 5, 15	3bitri 297	1 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)

Syntax hints:   ↔ wb 206   = wceq 1542   ∩ cin 3902  ∅c0 4287  dom cdm 5632  ran crn 5633   ↾ cres 5634   ∘ ccom 5636  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644

This theorem is referenced by:  coemptyd  14914  wrdpmtrlast  33186  diophrw  43113  relexpnul  44031