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Theorem arwhoma 14912
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 14910 . . . . . 6  |-  A  = 
U. ran  H
43eleq2i 2506 . . . . 5  |-  ( F  e.  A  <->  F  e.  U.
ran  H )
54biimpi 194 . . . 4  |-  ( F  e.  A  ->  F  e.  U. ran  H )
6 eqid 2442 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
71arwrcl 14911 . . . . . 6  |-  ( F  e.  A  ->  C  e.  Cat )
82, 6, 7homaf 14897 . . . . 5  |-  ( F  e.  A  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
9 ffn 5558 . . . . 5  |-  ( H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V )  ->  H  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
10 fnunirn 5969 . . . . 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 20 . . . 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 5690 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
14 df-ov 6093 . . . . . 6  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1513, 14syl6eqr 2492 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
1615eleq2d 2509 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F  e.  ( H `  z
)  <->  F  e.  (
x H y ) ) )
1716rexxp 4981 . . 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 14907 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (domA `  F )  =  x )
212homacd 14908 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (coda `  F
)  =  y )
2220, 21oveq12d 6108 . . . . 5  |-  ( F  e.  ( x H y )  ->  (
(domA `  F ) H (coda `  F ) )  =  ( x H y ) )
2319, 22eleqtrrd 2519 . . . 4  |-  ( F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2423rexlimivw 2836 . . 3  |-  ( E. y  e.  ( Base `  C ) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2524rexlimivw 2836 . 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 16 1  |-  ( F  e.  A  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   E.wrex 2715   _Vcvv 2971   ~Pcpw 3859   <.cop 3882   U.cuni 4090    X. cxp 4837   ran crn 4840    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090   Basecbs 14173  domAcdoma 14887  codaccoda 14888  Natcarw 14889  Homachoma 14890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-1st 6576  df-2nd 6577  df-doma 14891  df-coda 14892  df-homa 14893  df-arw 14894
This theorem is referenced by:  arwdm  14914  arwcd  14915  arwhom  14918  arwdmcd  14919  coapm  14938
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