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Theorem coeq0i 35595
Description: coeq0 5344 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 5735 . . . . . 6  |-  ( B : E --> F  ->  ran  B  C_  F )
213ad2ant2 1030 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ran  B 
C_  F )
3 sslin 3658 . . . . 5  |-  ( ran 
B  C_  F  ->  ( dom  A  i^i  ran  B )  C_  ( dom  A  i^i  F ) )
42, 3syl 17 . . . 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 5733 . . . . . . 7  |-  ( A : C --> D  ->  dom  A  =  C )
653ad2ant1 1029 . . . . . 6  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  dom  A  =  C )
76ineq1d 3633 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  ( C  i^i  F ) )
8 simp3 1010 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( C  i^i  F )  =  (/) )
97, 8eqtrd 2485 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  (/) )
104, 9sseqtrd 3468 . . 3  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  (/) )
11 ss0 3765 . . 3  |-  ( ( dom  A  i^i  ran  B )  C_  (/)  ->  ( dom  A  i^i  ran  B
)  =  (/) )
1210, 11syl 17 . 2  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  =  (/) )
13 coeq0 5344 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
1412, 13sylibr 216 1  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 985    = wceq 1444    i^i cin 3403    C_ wss 3404   (/)c0 3731   dom cdm 4834   ran crn 4835    o. ccom 4838   -->wf 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-fn 5585  df-f 5586
This theorem is referenced by:  diophren  35656
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