MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homfeq Structured version   Unicode version

Theorem homfeq 13951
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 2443 . . . . 5  |-  ( ph  ->  ( x H y )  =  ( x H y ) )
31, 1, 2mpt2eq123dv 6165 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) ) )
4 eqid 2442 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
5 eqid 2442 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 homfeq.h . . . . 5  |-  H  =  (  Hom  `  C
)
74, 5, 6homffval 13948 . . . 4  |-  (  Homf  `  C )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) )
83, 7syl6reqr 2493 . . 3  |-  ( ph  ->  (  Homf 
`  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
9 homfeq.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  D ) )
10 eqidd 2443 . . . . 5  |-  ( ph  ->  ( x J y )  =  ( x J y ) )
119, 9, 10mpt2eq123dv 6165 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) ) )
12 eqid 2442 . . . . 5  |-  (  Homf  `  D )  =  (  Homf 
`  D )
13 eqid 2442 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
14 homfeq.j . . . . 5  |-  J  =  (  Hom  `  D
)
1512, 13, 14homffval 13948 . . . 4  |-  (  Homf  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) )
1611, 15syl6reqr 2493 . . 3  |-  ( ph  ->  (  Homf 
`  D )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) ) )
178, 16eqeq12d 2456 . 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 6135 . . . 4  |-  ( x H y )  e. 
_V
1918rgen2w 2780 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x H y )  e. 
_V
20 mpt22eqb 6208 . . 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 5 . 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 254 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 178    = wceq 1653    e. wcel 1727   A.wral 2711   _Vcvv 2962   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   Basecbs 13500    Hom chom 13571    Homf chomf 13922
This theorem is referenced by:  homfeqd  13952  fullresc  14079  resssetc  14278  resscatc  14291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-homf 13926
  Copyright terms: Public domain W3C validator