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Theorem homacd 13873
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 13857 . . . 4  |- coda  =  ( 2nd  o. 
1st )
21fveq1i 5526 . . 3  |-  (coda `  F
)  =  ( ( 2nd  o.  1st ) `  F )
3 fo1st 6139 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5451 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 8 . . . 4  |-  1st : _V
--> _V
6 elex 2796 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5596 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
85, 6, 7sylancr 644 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
92, 8syl5eq 2327 . 2  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  ( 2nd `  ( 1st `  F
) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 13868 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6169 . . . . 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 13869 . . . 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 5529 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  ( 1st `  F
) )  =  ( 2nd `  <. X ,  Y >. ) )
17 eqid 2283 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 13867 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op2ndg 6133 . . 3  |-  ( ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2018, 19syl 15 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
219, 16, 203eqtrd 2319 1  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    o. ccom 4693   Rel wrel 4694   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148  codaccoda 13853  Homachoma 13855
This theorem is referenced by:  arwhoma  13877  idacd  13894  homdmcoa  13899  coaval  13900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-coda 13857  df-homa 13858
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