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Theorem arwhoma 13970
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 13968 . . . . . 6  |-  A  = 
U. ran  H
43eleq2i 2422 . . . . 5  |-  ( F  e.  A  <->  F  e.  U.
ran  H )
54biimpi 186 . . . 4  |-  ( F  e.  A  ->  F  e.  U. ran  H )
6 eqid 2358 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
71arwrcl 13969 . . . . . 6  |-  ( F  e.  A  ->  C  e.  Cat )
82, 6, 7homaf 13955 . . . . 5  |-  ( F  e.  A  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
9 ffn 5469 . . . . 5  |-  ( H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V )  ->  H  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
10 fnunirn 5862 . . . . 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 5605 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
14 df-ov 5945 . . . . . 6  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1513, 14syl6eqr 2408 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
1615eleq2d 2425 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F  e.  ( H `  z
)  <->  F  e.  (
x H y ) ) )
1716rexxp 4907 . . 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 13965 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (domA `  F )  =  x )
212homacd 13966 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (coda `  F
)  =  y )
2220, 21oveq12d 5960 . . . . 5  |-  ( F  e.  ( x H y )  ->  (
(domA `  F ) H (coda `  F ) )  =  ( x H y ) )
2319, 22eleqtrrd 2435 . . . 4  |-  ( F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2423rexlimivw 2739 . . 3  |-  ( E. y  e.  ( Base `  C ) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2524rexlimivw 2739 . 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 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   ~Pcpw 3701   <.cop 3719   U.cuni 3906    X. cxp 4766   ran crn 4769    Fn wfn 5329   -->wf 5330   ` cfv 5334  (class class class)co 5942   Basecbs 13239  domAcdoma 13945  codaccoda 13946  Natcarw 13947  Homachoma 13948
This theorem is referenced by:  arwdm  13972  arwcd  13973  arwhom  13976  arwdmcd  13977  coapm  13996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-1st 6206  df-2nd 6207  df-doma 13949  df-coda 13950  df-homa 13951  df-arw 13952
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