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Theorem homfeq 13696
Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
homfeq.h  |-  H  =  (  Hom  `  C
)
homfeq.j  |-  J  =  (  Hom  `  D
)
homfeq.1  |-  ( ph  ->  B  =  ( Base `  C ) )
homfeq.2  |-  ( ph  ->  B  =  ( Base `  D ) )
Assertion
Ref Expression
homfeq  |-  ( ph  ->  ( (  Homf  `  C )  =  (  Homf  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
Distinct variable groups:    x, y, B    x, C, y    x, D, y    x, H, y    ph, x, y    x, J, y

Proof of Theorem homfeq
StepHypRef Expression
1 homfeq.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  C ) )
2 eqidd 2359 . . . . 5  |-  ( ph  ->  ( x H y )  =  ( x H y ) )
31, 1, 2mpt2eq123dv 5997 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) ) )
4 eqid 2358 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
5 eqid 2358 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 homfeq.h . . . . 5  |-  H  =  (  Hom  `  C
)
74, 5, 6homffval 13693 . . . 4  |-  (  Homf  `  C )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) )
83, 7syl6reqr 2409 . . 3  |-  ( ph  ->  (  Homf 
`  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
9 homfeq.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  D ) )
10 eqidd 2359 . . . . 5  |-  ( ph  ->  ( x J y )  =  ( x J y ) )
119, 9, 10mpt2eq123dv 5997 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) ) )
12 eqid 2358 . . . . 5  |-  (  Homf  `  D )  =  (  Homf 
`  D )
13 eqid 2358 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
14 homfeq.j . . . . 5  |-  J  =  (  Hom  `  D
)
1512, 13, 14homffval 13693 . . . 4  |-  (  Homf  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) )
1611, 15syl6reqr 2409 . . 3  |-  ( ph  ->  (  Homf 
`  D )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) ) )
178, 16eqeq12d 2372 . 2  |-  ( ph  ->  ( (  Homf  `  C )  =  (  Homf  `  D )  <-> 
( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  B , 
y  e.  B  |->  ( x J y ) ) ) )
18 ovex 5970 . . . 4  |-  ( x H y )  e. 
_V
1918rgen2w 2687 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x H y )  e. 
_V
20 mpt22eqb 6040 . . 3  |-  ( A. x  e.  B  A. y  e.  B  (
x H y )  e.  _V  ->  (
( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  B , 
y  e.  B  |->  ( x J y ) )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
2119, 20ax-mp 8 . 2  |-  ( ( x  e.  B , 
y  e.  B  |->  ( x H y ) )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) )
2217, 21syl6bb 252 1  |-  ( ph  ->  ( (  Homf  `  C )  =  (  Homf  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   Basecbs 13245    Hom chom 13316    Homf chomf 13667
This theorem is referenced by:  homfeqd  13697  fullresc  13824  resssetc  14023  resscatc  14036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-homf 13671
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