Theorem coemptyd 15015

Description: Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
coemptyd.1	⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)

Assertion

Ref	Expression
coemptyd	⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Proof of Theorem coemptyd

Step	Hyp	Ref	Expression
1		coemptyd.1	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)
2		coeq0 6277	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
3	1, 2	sylibr 234	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3962  ∅c0 4339  dom cdm 5689  ran crn 5690   ∘ ccom 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701

This theorem is referenced by:  xptrrel  15016  cosnopne  32709  coeq0i  42741  conrel1d  43653  conrel2d  43654  clsneibex  44092  neicvgbex  44102