MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homadm Unicode version

Theorem homadm 14082
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 14066 . . . 4  |- domA 
=  ( 1st  o.  1st )
21fveq1i 5633 . . 3  |-  (domA `  F )  =  ( ( 1st 
o.  1st ) `  F
)
3 fo1st 6266 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5557 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 8 . . . 4  |-  1st : _V
--> _V
6 elex 2881 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5703 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
85, 6, 7sylancr 644 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
92, 8syl5eq 2410 . 2  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  ( 1st `  ( 1st `  F ) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 14078 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6296 . . . . 5  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1311, 12mpan 651 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1410homa1 14079 . . . 4  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 1st `  F )  =  <. X ,  Y >. )
1513, 14syl 15 . . 3  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F )  = 
<. X ,  Y >. )
1615fveq2d 5636 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  ( 1st `  F
) )  =  ( 1st `  <. X ,  Y >. ) )
17 eqid 2366 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 14077 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op1stg 6259 . . 3  |-  ( ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2018, 19syl 15 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  <. X ,  Y >. )  =  X )
219, 16, 203eqtrd 2402 1  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   <.cop 3732   class class class wbr 4125    o. ccom 4796   Rel wrel 4797   -->wf 5354   -onto->wfo 5356   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   Basecbs 13356  domAcdoma 14062  Homachoma 14065
This theorem is referenced by:  arwhoma  14087  idadm  14103  homdmcoa  14109  coaval  14110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-1st 6249  df-2nd 6250  df-doma 14066  df-homa 14068
  Copyright terms: Public domain W3C validator