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Theorem homaval 15233
Description: Value of the disjointified hom-set function. (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 )
Assertion
Ref Expression
homaval  |-  ( ph  ->  ( X H Y )  =  ( {
<. X ,  Y >. }  X.  ( X J Y ) ) )

Proof of Theorem homaval
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ov 6298 . 2  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
2 homarcl.h . . . 4  |-  H  =  (Homa
`  C )
3 homafval.b . . . 4  |-  B  =  ( Base `  C
)
4 homafval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 homaval.j . . . 4  |-  J  =  ( Hom  `  C
)
62, 3, 4, 5homafval 15231 . . 3  |-  ( ph  ->  H  =  ( z  e.  ( B  X.  B )  |->  ( { z }  X.  ( J `  z )
) ) )
7 simpr 461 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
87sneqd 4045 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  { z }  =  { <. X ,  Y >. } )
97fveq2d 5876 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( J `  <. X ,  Y >. ) )
10 df-ov 6298 . . . . 5  |-  ( X J Y )  =  ( J `  <. X ,  Y >. )
119, 10syl6eqr 2526 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( X J Y ) )
128, 11xpeq12d 5030 . . 3  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( { z }  X.  ( J `
 z ) )  =  ( { <. X ,  Y >. }  X.  ( X J Y ) ) )
13 homaval.x . . . 4  |-  ( ph  ->  X  e.  B )
14 homaval.y . . . 4  |-  ( ph  ->  Y  e.  B )
15 opelxpi 5037 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
1613, 14, 15syl2anc 661 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
17 snex 4694 . . . . 5  |-  { <. X ,  Y >. }  e.  _V
18 ovex 6320 . . . . 5  |-  ( X J Y )  e. 
_V
1917, 18xpex 6599 . . . 4  |-  ( {
<. X ,  Y >. }  X.  ( X J Y ) )  e. 
_V
2019a1i 11 . . 3  |-  ( ph  ->  ( { <. X ,  Y >. }  X.  ( X J Y ) )  e.  _V )
216, 12, 16, 20fvmptd 5962 . 2  |-  ( ph  ->  ( H `  <. X ,  Y >. )  =  ( { <. X ,  Y >. }  X.  ( X J Y ) ) )
221, 21syl5eq 2520 1  |-  ( ph  ->  ( X H Y )  =  ( {
<. X ,  Y >. }  X.  ( X J Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033   <.cop 4039    X. cxp 5003   ` cfv 5594  (class class class)co 6295   Basecbs 14507   Hom chom 14583   Catccat 14936  Homachoma 15225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-homa 15228
This theorem is referenced by:  elhoma  15234
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