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Theorem homacd 15243
Description: The codomain 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
homacd  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )

Proof of Theorem homacd
StepHypRef Expression
1 df-coda 15227 . . . 4  |- coda  =  ( 2nd  o. 
1st )
21fveq1i 5873 . . 3  |-  (coda `  F
)  =  ( ( 2nd  o.  1st ) `  F )
3 fo1st 6815 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5801 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . 4  |-  1st : _V
--> _V
6 elex 3127 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5951 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
85, 6, 7sylancr 663 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
92, 8syl5eq 2520 . 2  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  ( 2nd `  ( 1st `  F
) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 15238 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6844 . . . . 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 15239 . . . 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 5876 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  ( 1st `  F
) )  =  ( 2nd `  <. X ,  Y >. ) )
17 eqid 2467 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 15237 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op2ndg 6808 . . 3  |-  ( ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2018, 19syl 16 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
219, 16, 203eqtrd 2512 1  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039   class class class wbr 4453    o. ccom 5009   Rel wrel 5010   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   Basecbs 14507  codaccoda 15223  Homachoma 15225
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-1st 6795  df-2nd 6796  df-coda 15227  df-homa 15228
This theorem is referenced by:  arwhoma  15247  idacd  15264  homdmcoa  15269  coaval  15270
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