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Theorem elhomai 14115
Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homarcl.h  |-  H  =  (Homa
`  C )
homafval.b  |-  B  =  ( Base `  C
)
homafval.c  |-  ( ph  ->  C  e.  Cat )
homaval.j  |-  J  =  (  Hom  `  C
)
homaval.x  |-  ( ph  ->  X  e.  B )
homaval.y  |-  ( ph  ->  Y  e.  B )
elhomai.f  |-  ( ph  ->  F  e.  ( X J Y ) )
Assertion
Ref Expression
elhomai  |-  ( ph  -> 
<. X ,  Y >. ( X H Y ) F )

Proof of Theorem elhomai
StepHypRef Expression
1 eqidd 2388 . 2  |-  ( ph  -> 
<. X ,  Y >.  = 
<. X ,  Y >. )
2 elhomai.f . 2  |-  ( ph  ->  F  e.  ( X J Y ) )
3 homarcl.h . . 3  |-  H  =  (Homa
`  C )
4 homafval.b . . 3  |-  B  =  ( Base `  C
)
5 homafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
6 homaval.j . . 3  |-  J  =  (  Hom  `  C
)
7 homaval.x . . 3  |-  ( ph  ->  X  e.  B )
8 homaval.y . . 3  |-  ( ph  ->  Y  e.  B )
93, 4, 5, 6, 7, 8elhoma 14114 . 2  |-  ( ph  ->  ( <. X ,  Y >. ( X H Y ) F  <->  ( <. X ,  Y >.  =  <. X ,  Y >.  /\  F  e.  ( X J Y ) ) ) )
101, 2, 9mpbir2and 889 1  |-  ( ph  -> 
<. X ,  Y >. ( X H Y ) F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3760   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396    Hom chom 13467   Catccat 13816  Homachoma 14105
This theorem is referenced by:  elhomai2  14116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-homa 14108
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