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Theorem coeq0i 30279
Description: coeq0 5509 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 5730 . . . . . 6  |-  ( B : E --> F  ->  ran  B  C_  F )
213ad2ant2 1013 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ran  B 
C_  F )
3 sslin 3719 . . . . 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 5728 . . . . . . 7  |-  ( A : C --> D  ->  dom  A  =  C )
653ad2ant1 1012 . . . . . 6  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  dom  A  =  C )
76ineq1d 3694 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  ( C  i^i  F ) )
8 simp3 993 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( C  i^i  F )  =  (/) )
97, 8eqtrd 2503 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  (/) )
104, 9sseqtrd 3535 . . 3  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  (/) )
11 ss0 3811 . . 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 5509 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
1412, 13sylibr 212 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 968    = wceq 1374    i^i cin 3470    C_ wss 3471   (/)c0 3780   dom cdm 4994   ran crn 4995    o. ccom 4998   -->wf 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-fn 5584  df-f 5585
This theorem is referenced by:  diophren  30340
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