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Theorem arwhoma 15648
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 15646 . . . . . 6  |-  A  = 
U. ran  H
43eleq2i 2480 . . . . 5  |-  ( F  e.  A  <->  F  e.  U.
ran  H )
54biimpi 194 . . . 4  |-  ( F  e.  A  ->  F  e.  U. ran  H )
6 eqid 2402 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
71arwrcl 15647 . . . . . 6  |-  ( F  e.  A  ->  C  e.  Cat )
82, 6, 7homaf 15633 . . . . 5  |-  ( F  e.  A  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
9 ffn 5714 . . . . 5  |-  ( H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V )  ->  H  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
10 fnunirn 6146 . . . . 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 210 . . 3  |-  ( F  e.  A  ->  E. z  e.  ( ( Base `  C
)  X.  ( Base `  C ) ) F  e.  ( H `  z ) )
13 fveq2 5849 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
14 df-ov 6281 . . . . . 6  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1513, 14syl6eqr 2461 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
1615eleq2d 2472 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F  e.  ( H `  z
)  <->  F  e.  (
x H y ) ) )
1716rexxp 4966 . . 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 196 . 2  |-  ( F  e.  A  ->  E. x  e.  ( Base `  C
) E. y  e.  ( Base `  C
) F  e.  ( x H y ) )
19 id 22 . . . . 5  |-  ( F  e.  ( x H y )  ->  F  e.  ( x H y ) )
202homadm 15643 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (domA `  F )  =  x )
212homacd 15644 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (coda `  F
)  =  y )
2220, 21oveq12d 6296 . . . . 5  |-  ( F  e.  ( x H y )  ->  (
(domA `  F ) H (coda `  F ) )  =  ( x H y ) )
2319, 22eleqtrrd 2493 . . . 4  |-  ( F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2423rexlimivw 2893 . . 3  |-  ( E. y  e.  ( Base `  C ) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2524rexlimivw 2893 . 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 17 1  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059   ~Pcpw 3955   <.cop 3978   U.cuni 4191    X. cxp 4821   ran crn 4824    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   Basecbs 14841  domAcdoma 15623  codaccoda 15624  Natcarw 15625  Homachoma 15626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-1st 6784  df-2nd 6785  df-doma 15627  df-coda 15628  df-homa 15629  df-arw 15630
This theorem is referenced by:  arwdm  15650  arwcd  15651  arwhom  15654  arwdmcd  15655  coapm  15674
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