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Theorem coemptyd 13055
Description: Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
coemptyd.1  |-  ( ph  ->  ( dom  A  i^i  ran 
B )  =  (/) )
Assertion
Ref Expression
coemptyd  |-  ( ph  ->  ( A  o.  B
)  =  (/) )

Proof of Theorem coemptyd
StepHypRef Expression
1 coemptyd.1 . 2  |-  ( ph  ->  ( dom  A  i^i  ran 
B )  =  (/) )
2 coeq0 5347 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
31, 2sylibr 216 1  |-  ( ph  ->  ( A  o.  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    i^i cin 3405   (/)c0 3733   dom cdm 4837   ran crn 4838    o. ccom 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849
This theorem is referenced by:  xptrrel  13056  conrel1d  36267  conrel2d  36268
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