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Theorem homacd 15031
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 15015 . . . 4  |- coda  =  ( 2nd  o. 
1st )
21fveq1i 5803 . . 3  |-  (coda `  F
)  =  ( ( 2nd  o.  1st ) `  F )
3 fo1st 6709 . . . . 5  |-  1st : _V -onto-> _V
4 fof 5731 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
53, 4ax-mp 5 . . . 4  |-  1st : _V
--> _V
6 elex 3087 . . . 4  |-  ( F  e.  ( X H Y )  ->  F  e.  _V )
7 fvco3 5880 . . . 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 2507 . 2  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  ( 2nd `  ( 1st `  F
) ) )
10 homahom.h . . . . . 6  |-  H  =  (Homa
`  C )
1110homarel 15026 . . . . 5  |-  Rel  ( X H Y )
12 1st2ndbr 6736 . . . . 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 15027 . . . 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 5806 . 2  |-  ( F  e.  ( X H Y )  ->  ( 2nd `  ( 1st `  F
) )  =  ( 2nd `  <. X ,  Y >. ) )
17 eqid 2454 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
1810, 17homarcl2 15025 . . 3  |-  ( F  e.  ( X H Y )  ->  ( X  e.  ( Base `  C )  /\  Y  e.  ( Base `  C
) ) )
19 op2ndg 6703 . . 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 2499 1  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994   class class class wbr 4403    o. ccom 4955   Rel wrel 4956   -->wf 5525   -onto->wfo 5527   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689   Basecbs 14295  codaccoda 15011  Homachoma 15013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-1st 6690  df-2nd 6691  df-coda 15015  df-homa 15016
This theorem is referenced by:  arwhoma  15035  idacd  15052  homdmcoa  15057  coaval  15058
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