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Theorem coeq0i 30848
Description: coeq0 5522 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 5743 . . . . . 6  |-  ( B : E --> F  ->  ran  B  C_  F )
213ad2ant2 1018 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ran  B 
C_  F )
3 sslin 3720 . . . . 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 5741 . . . . . . 7  |-  ( A : C --> D  ->  dom  A  =  C )
653ad2ant1 1017 . . . . . 6  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  dom  A  =  C )
76ineq1d 3695 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  ( C  i^i  F ) )
8 simp3 998 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( C  i^i  F )  =  (/) )
97, 8eqtrd 2498 . . . 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 3825 . . 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 5522 . 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 973    = wceq 1395    i^i cin 3470    C_ wss 3471   (/)c0 3793   dom cdm 5008   ran crn 5009    o. ccom 5012   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-fn 5597  df-f 5598
This theorem is referenced by:  diophren  30909
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