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Theorem arwhoma 15221
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 15219 . . . . . 6  |-  A  = 
U. ran  H
43eleq2i 2540 . . . . 5  |-  ( F  e.  A  <->  F  e.  U.
ran  H )
54biimpi 194 . . . 4  |-  ( F  e.  A  ->  F  e.  U. ran  H )
6 eqid 2462 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
71arwrcl 15220 . . . . . 6  |-  ( F  e.  A  ->  C  e.  Cat )
82, 6, 7homaf 15206 . . . . 5  |-  ( F  e.  A  ->  H : ( ( Base `  C )  X.  ( Base `  C ) ) --> ~P ( ( (
Base `  C )  X.  ( Base `  C
) )  X.  _V ) )
9 ffn 5724 . . . . 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 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 5859 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( H `  <. x ,  y >. )
)
14 df-ov 6280 . . . . . 6  |-  ( x H y )  =  ( H `  <. x ,  y >. )
1513, 14syl6eqr 2521 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( H `  z )  =  ( x H y ) )
1615eleq2d 2532 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( F  e.  ( H `  z
)  <->  F  e.  (
x H y ) ) )
1716rexxp 5138 . . 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 15216 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (domA `  F )  =  x )
212homacd 15217 . . . . . 6  |-  ( F  e.  ( x H y )  ->  (coda `  F
)  =  y )
2220, 21oveq12d 6295 . . . . 5  |-  ( F  e.  ( x H y )  ->  (
(domA `  F ) H (coda `  F ) )  =  ( x H y ) )
2319, 22eleqtrrd 2553 . . . 4  |-  ( F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2423rexlimivw 2947 . . 3  |-  ( E. y  e.  ( Base `  C ) F  e.  ( x H y )  ->  F  e.  ( (domA `  F ) H (coda `  F ) ) )
2524rexlimivw 2947 . 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 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108   ~Pcpw 4005   <.cop 4028   U.cuni 4240    X. cxp 4992   ran crn 4995    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   Basecbs 14481  domAcdoma 15196  codaccoda 15197  Natcarw 15198  Homachoma 15199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-1st 6776  df-2nd 6777  df-doma 15200  df-coda 15201  df-homa 15202  df-arw 15203
This theorem is referenced by:  arwdm  15223  arwcd  15224  arwhom  15227  arwdmcd  15228  coapm  15247
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