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Theorem homadm 15225
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 15209 . . . 4  |- domA 
=  ( 1st  o.  1st )
21fveq1i 5867 . . 3  |-  (domA `  F )  =  ( ( 1st 
o.  1st ) `  F
)
3 fo1st 6804 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5795 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . 4  |-  1st : _V
--> _V
6 elex 3122 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5944 . . . 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 2520 . 2  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  ( 1st `  ( 1st `  F ) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 15221 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6833 . . . . 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 15222 . . . 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 5870 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  ( 1st `  F
) )  =  ( 1st `  <. X ,  Y >. ) )
17 eqid 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 15220 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op1stg 6796 . . 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 2512 1  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447    o. ccom 5003   Rel wrel 5004   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   Basecbs 14490  domAcdoma 15205  Homachoma 15208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-1st 6784  df-2nd 6785  df-doma 15209  df-homa 15211
This theorem is referenced by:  arwhoma  15230  idadm  15246  homdmcoa  15252  coaval  15253
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