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Theorem homadm 14158
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 14142 . . . 4  |- domA 
=  ( 1st  o.  1st )
21fveq1i 5696 . . 3  |-  (domA `  F )  =  ( ( 1st 
o.  1st ) `  F
)
3 fo1st 6333 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5620 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 8 . . . 4  |-  1st : _V
--> _V
6 elex 2932 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5767 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
85, 6, 7sylancr 645 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 1st  o.  1st ) `  F )  =  ( 1st `  ( 1st `  F ) ) )
92, 8syl5eq 2456 . 2  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  ( 1st `  ( 1st `  F ) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 14154 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6363 . . . . 5  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1311, 12mpan 652 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
1410homa1 14155 . . . 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 5699 . 2  |-  ( F  e.  ( X H Y )  ->  ( 1st `  ( 1st `  F
) )  =  ( 1st `  <. X ,  Y >. ) )
17 eqid 2412 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 14153 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op1stg 6326 . . 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 2448 1  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   <.cop 3785   class class class wbr 4180    o. ccom 4849   Rel wrel 4850   -->wf 5417   -onto->wfo 5419   ` cfv 5421  (class class class)co 6048   1stc1st 6314   2ndc2nd 6315   Basecbs 13432  domAcdoma 14138  Homachoma 14141
This theorem is referenced by:  arwhoma  14163  idadm  14179  homdmcoa  14185  coaval  14186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-1st 6316  df-2nd 6317  df-doma 14142  df-homa 14144
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