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Theorem homadm 15886
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 15870 . . . 4  |- domA 
=  ( 1st  o.  1st )
21fveq1i 5882 . . 3  |-  (domA `  F )  =  ( ( 1st 
o.  1st ) `  F
)
3 fo1st 6827 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5810 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . 4  |-  1st : _V
--> _V
6 elex 3096 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5958 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
85, 6, 7sylancr 667 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
92, 8syl5eq 2482 . 2  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  ( 1st `  ( 1st `  F ) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 15882 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6856 . . . . 5  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1311, 12mpan 674 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1410homa1 15883 . . . 4  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 1st `  F )  =  <. X ,  Y >. )
1513, 14syl 17 . . 3  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F )  = 
<. X ,  Y >. )
1615fveq2d 5885 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  ( 1st `  F
) )  =  ( 1st `  <. X ,  Y >. ) )
17 eqid 2429 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 15881 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op1stg 6819 . . 3  |-  ( ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2018, 19syl 17 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  <. X ,  Y >. )  =  X )
219, 16, 203eqtrd 2474 1  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   <.cop 4008   class class class wbr 4426    o. ccom 4858   Rel wrel 4859   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2ndc2nd 6806   Basecbs 15084  domAcdoma 15866  Homachoma 15869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-1st 6807  df-2nd 6808  df-doma 15870  df-homa 15872
This theorem is referenced by:  arwhoma  15891  idadm  15907  homdmcoa  15913  coaval  15914
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