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Theorem homaval 14178
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 6076 . 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 14176 . . 3  |-  ( ph  ->  H  =  ( z  e.  ( B  X.  B )  |->  ( { z }  X.  ( J `  z )
) ) )
7 simpr 448 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  z  =  <. X ,  Y >. )
87sneqd 3819 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  { z }  =  { <. X ,  Y >. } )
97fveq2d 5724 . . . . 5  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( J `  <. X ,  Y >. ) )
10 df-ov 6076 . . . . 5  |-  ( X J Y )  =  ( J `  <. X ,  Y >. )
119, 10syl6eqr 2485 . . . 4  |-  ( (
ph  /\  z  =  <. X ,  Y >. )  ->  ( J `  z )  =  ( X J Y ) )
128, 11xpeq12d 4895 . . 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 4902 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
1613, 14, 15syl2anc 643 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
17 snex 4397 . . . . 5  |-  { <. X ,  Y >. }  e.  _V
18 ovex 6098 . . . . 5  |-  ( X J Y )  e. 
_V
1917, 18xpex 4982 . . . 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 5802 . 2  |-  ( ph  ->  ( H `  <. X ,  Y >. )  =  ( { <. X ,  Y >. }  X.  ( X J Y ) ) )
221, 21syl5eq 2479 1  |-  ( ph  ->  ( X H Y )  =  ( {
<. X ,  Y >. }  X.  ( X J Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809    X. cxp 4868   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532   Catccat 13881  Homachoma 14170
This theorem is referenced by:  elhoma  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 14173
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