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Theorem elhomai 14178
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 2436 . 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 14177 . 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 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459    Hom chom 13530   Catccat 13879  Homachoma 14168
This theorem is referenced by:  elhomai2  14179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-homa 14171
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