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Theorem homacd 13889
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 13873 . . . 4  |- coda  =  ( 2nd  o. 
1st )
21fveq1i 5542 . . 3  |-  (coda `  F
)  =  ( ( 2nd  o.  1st ) `  F )
3 fo1st 6155 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5467 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 8 . . . 4  |-  1st : _V
--> _V
6 elex 2809 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5612 . . . 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 2340 . 2  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  ( 2nd `  ( 1st `  F
) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 13884 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6185 . . . . 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 13885 . . . 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 5545 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  ( 1st `  F
) )  =  ( 2nd `  <. X ,  Y >. ) )
17 eqid 2296 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 13883 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op2ndg 6149 . . 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 2332 1  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    o. ccom 4709   Rel wrel 4710   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164  codaccoda 13869  Homachoma 13871
This theorem is referenced by:  arwhoma  13893  idacd  13910  homdmcoa  13915  coaval  13916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-coda 13873  df-homa 13874
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