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Theorem homaval 14898
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 6093 . 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 14896 . . 3  |-  ( ph  ->  H  =  ( z  e.  ( B  X.  B )  |->  ( { z }  X.  ( J `  z )
) ) )
7 simpr 461 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
87sneqd 3888 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  { z }  =  { <. X ,  Y >. } )
97fveq2d 5694 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( J `  <. X ,  Y >. ) )
10 df-ov 6093 . . . . 5  |-  ( X J Y )  =  ( J `  <. X ,  Y >. )
119, 10syl6eqr 2492 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( X J Y ) )
128, 11xpeq12d 4864 . . 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 4870 . . . 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 4532 . . . . 5  |-  { <. X ,  Y >. }  e.  _V
18 ovex 6115 . . . . 5  |-  ( X J Y )  e. 
_V
1917, 18xpex 6507 . . . 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 5778 . 2  |-  ( ph  ->  ( H `  <. X ,  Y >. )  =  ( { <. X ,  Y >. }  X.  ( X J Y ) ) )
221, 21syl5eq 2486 1  |-  ( ph  ->  ( X H Y )  =  ( {
<. X ,  Y >. }  X.  ( X J Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2971   {csn 3876   <.cop 3882    X. cxp 4837   ` cfv 5417  (class class class)co 6090   Basecbs 14173   Hom chom 14248   Catccat 14601  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-homa 14893
This theorem is referenced by:  elhoma  14899
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