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Theorem arwhoma 13877
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a  |-  A  =  (Nat `  C )
arwhoma.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
arwhoma  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )

Proof of Theorem arwhoma
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 arwrcl.a . . . . . . 7  |-  A  =  (Nat `  C )
2 arwhoma.h . . . . . . 7  |-  H  =  (Homa
`  C )
31, 2arwval 13875 . . . . . 6  |-  A  = 
U. ran  H
43eleq2i 2347 . . . . 5  |-  ( F  e.  A  <->  F  e.  U.
ran  H )
54biimpi 186 . . . 4  |-  ( F  e.  A  ->  F  e.  U. ran  H )
6 eqid 2283 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
71arwrcl 13876 . . . . . 6  |-  ( F  e.  A  ->  C  e.  Cat )
82, 6, 7homaf 13862 . . . . 5  |-  ( F  e.  A  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
9 ffn 5389 . . . . 5  |-  ( H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V )  ->  H  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
10 fnunirn 5778 . . . . 5  |-  ( H  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  ->  ( F  e. 
U. ran  H  <->  E. z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) F  e.  ( H `  z ) ) )
118, 9, 103syl 18 . . . 4  |-  ( F  e.  A  ->  ( F  e.  U. ran  H  <->  E. z  e.  ( (
Base `  C )  X.  ( Base `  C
) ) F  e.  ( H `  z
) ) )
125, 11mpbid 201 . . 3  |-  ( F  e.  A  ->  E. z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) F  e.  ( H `  z ) )
13 fveq2 5525 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
14 df-ov 5861 . . . . . 6  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1513, 14syl6eqr 2333 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
1615eleq2d 2350 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F  e.  ( H `  z
)  <->  F  e.  (
x H y ) ) )
1716rexxp 4828 . . 3  |-  ( E. z  e.  ( (
Base `  C )  X.  ( Base `  C
) ) F  e.  ( H `  z
)  <->  E. x  e.  (
Base `  C ) E. y  e.  ( Base `  C ) F  e.  ( x H y ) )
1812, 17sylib 188 . 2  |-  ( F  e.  A  ->  E. x  e.  ( Base `  C
) E. y  e.  ( Base `  C
) F  e.  ( x H y ) )
19 id 19 . . . . 5  |-  ( F  e.  ( x H y )  ->  F  e.  ( x H y ) )
202homadm 13872 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (domA `  F )  =  x )
212homacd 13873 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (coda `  F
)  =  y )
2220, 21oveq12d 5876 . . . . 5  |-  ( F  e.  ( x H y )  ->  (
(domA `  F ) H (coda `  F ) )  =  ( x H y ) )
2319, 22eleqtrrd 2360 . . . 4  |-  ( F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2423rexlimivw 2663 . . 3  |-  ( E. y  e.  ( Base `  C ) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2524rexlimivw 2663 . 2  |-  ( E. x  e.  ( Base `  C ) E. y  e.  ( Base `  C
) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2618, 25syl 15 1  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   ~Pcpw 3625   <.cop 3643   U.cuni 3827    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  domAcdoma 13852  codaccoda 13853  Natcarw 13854  Homachoma 13855
This theorem is referenced by:  arwdm  13879  arwcd  13880  arwhom  13883  arwdmcd  13884  coapm  13903
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-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859
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