Theorem coemptyd 14935

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 6215	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
3	1, 2	sylibr 234	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3889  ∅c0 4274  dom cdm 5625  ran crn 5626   ∘ ccom 5629

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 5232  ax-pr 5371

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637

This theorem is referenced by:  xptrrel  14936  cosnopne  32785  coeq0i  43202  conrel1d  44111  conrel2d  44112  clsneibex  44550  neicvgbex  44560