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Theorem homadm 14913
Description: The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homadm  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )

Proof of Theorem homadm
StepHypRef Expression
1 df-doma 14897 . . . 4  |- domA 
=  ( 1st  o.  1st )
21fveq1i 5697 . . 3  |-  (domA `  F )  =  ( ( 1st 
o.  1st ) `  F
)
3 fo1st 6601 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5625 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . 4  |-  1st : _V
--> _V
6 elex 2986 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5773 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
85, 6, 7sylancr 663 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
92, 8syl5eq 2487 . 2  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  ( 1st `  ( 1st `  F ) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 14909 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6628 . . . . 5  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1311, 12mpan 670 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1410homa1 14910 . . . 4  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 1st `  F )  =  <. X ,  Y >. )
1513, 14syl 16 . . 3  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F )  = 
<. X ,  Y >. )
1615fveq2d 5700 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  ( 1st `  F
) )  =  ( 1st `  <. X ,  Y >. ) )
17 eqid 2443 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 14908 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op1stg 6594 . . 3  |-  ( ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2018, 19syl 16 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  <. X ,  Y >. )  =  X )
219, 16, 203eqtrd 2479 1  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888   class class class wbr 4297    o. ccom 4849   Rel wrel 4850   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Basecbs 14179  domAcdoma 14893  Homachoma 14896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-1st 6582  df-2nd 6583  df-doma 14897  df-homa 14899
This theorem is referenced by:  arwhoma  14918  idadm  14934  homdmcoa  14940  coaval  14941
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