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Theorem coeq0i 26811
Description: coeq0 26810 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
coeq0i  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )

Proof of Theorem coeq0i
StepHypRef Expression
1 frn 5597 . . . . . 6  |-  ( B : E --> F  ->  ran  B  C_  F )
213ad2ant2 979 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ran  B 
C_  F )
3 sslin 3567 . . . . 5  |-  ( ran 
B  C_  F  ->  ( dom  A  i^i  ran  B )  C_  ( dom  A  i^i  F ) )
42, 3syl 16 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  ( dom  A  i^i  F ) )
5 fdm 5595 . . . . . . 7  |-  ( A : C --> D  ->  dom  A  =  C )
653ad2ant1 978 . . . . . 6  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  dom  A  =  C )
76ineq1d 3541 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  ( C  i^i  F ) )
8 simp3 959 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( C  i^i  F )  =  (/) )
97, 8eqtrd 2468 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  (/) )
104, 9sseqtrd 3384 . . 3  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  (/) )
11 ss0 3658 . . 3  |-  ( ( dom  A  i^i  ran  B )  C_  (/)  ->  ( dom  A  i^i  ran  B
)  =  (/) )
1210, 11syl 16 . 2  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  =  (/) )
13 coeq0 26810 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
1412, 13sylibr 204 1  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    i^i cin 3319    C_ wss 3320   (/)c0 3628   dom cdm 4878   ran crn 4879    o. ccom 4882   -->wf 5450
This theorem is referenced by:  diophren  26874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-fn 5457  df-f 5458
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