Theorem coemptyd 14926

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 6255	. 2 ⊢ ((𝐴 ∘ 𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
3	1, 2	sylibr 233	1 ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3948  ∅c0 4323  dom cdm 5677  ran crn 5678   ∘ ccom 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689

This theorem is referenced by:  xptrrel  14927  cosnopne  31916  coeq0i  41491  conrel1d  42414  conrel2d  42415  clsneibex  42853  neicvgbex  42863