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

Theorem homacd 14196
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 14180 . . . 4  |- coda  =  ( 2nd  o. 
1st )
21fveq1i 5729 . . 3  |-  (coda `  F
)  =  ( ( 2nd  o.  1st ) `  F )
3 fo1st 6366 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5653 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 8 . . . 4  |-  1st : _V
--> _V
6 elex 2964 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5800 . . . 4  |-  ( ( 1st : _V --> _V  /\  F  e.  _V )  ->  ( ( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
85, 6, 7sylancr 645 . . 3  |-  ( F  e.  ( X H Y )  ->  (
( 2nd  o.  1st ) `  F )  =  ( 2nd `  ( 1st `  F ) ) )
92, 8syl5eq 2480 . 2  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  ( 2nd `  ( 1st `  F
) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 14191 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6396 . . . . 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 14192 . . . 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 5732 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  ( 1st `  F
) )  =  ( 2nd `  <. X ,  Y >. ) )
17 eqid 2436 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 14190 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op2ndg 6360 . . 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 2472 1  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212    o. ccom 4882   Rel wrel 4883   -->wf 5450   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Basecbs 13469  codaccoda 14176  Homachoma 14178
This theorem is referenced by:  arwhoma  14200  idacd  14217  homdmcoa  14222  coaval  14223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-coda 14180  df-homa 14181
  Copyright terms: Public domain W3C validator