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Theorem homfeq 13879
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 2409 . . . . 5  |-  ( ph  ->  ( x H y )  =  ( x H y ) )
31, 1, 2mpt2eq123dv 6099 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) ) )
4 eqid 2408 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  C )
5 eqid 2408 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 homfeq.h . . . . 5  |-  H  =  (  Hom  `  C
)
74, 5, 6homffval 13876 . . . 4  |-  (  Homf  `  C )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) )
83, 7syl6reqr 2459 . . 3  |-  ( ph  ->  (  Homf 
`  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
9 homfeq.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  D ) )
10 eqidd 2409 . . . . 5  |-  ( ph  ->  ( x J y )  =  ( x J y ) )
119, 9, 10mpt2eq123dv 6099 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) ) )
12 eqid 2408 . . . . 5  |-  (  Homf  `  D )  =  (  Homf 
`  D )
13 eqid 2408 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
14 homfeq.j . . . . 5  |-  J  =  (  Hom  `  D
)
1512, 13, 14homffval 13876 . . . 4  |-  (  Homf  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) )
1611, 15syl6reqr 2459 . . 3  |-  ( ph  ->  (  Homf 
`  D )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) ) )
178, 16eqeq12d 2422 . 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 6069 . . . 4  |-  ( x H y )  e. 
_V
1918rgen2w 2738 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x H y )  e. 
_V
20 mpt22eqb 6142 . . 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 253 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 177    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   Basecbs 13428    Hom chom 13499    Homf chomf 13850
This theorem is referenced by:  homfeqd  13880  fullresc  14007  resssetc  14206  resscatc  14219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-homf 13854
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