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Theorem homfeq 14638
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  ->  ( ( Hom f  `  C )  =  ( Hom f  `  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 2444 . . . . 5  |-  ( ph  ->  ( x H y )  =  ( x H y ) )
31, 1, 2mpt2eq123dv 6153 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) ) )
4 eqid 2443 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
5 eqid 2443 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 homfeq.h . . . . 5  |-  H  =  ( Hom  `  C
)
74, 5, 6homffval 14635 . . . 4  |-  ( Hom f  `  C )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) )
83, 7syl6reqr 2494 . . 3  |-  ( ph  ->  ( Hom f  `  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
9 homfeq.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  D ) )
10 eqidd 2444 . . . . 5  |-  ( ph  ->  ( x J y )  =  ( x J y ) )
119, 9, 10mpt2eq123dv 6153 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) ) )
12 eqid 2443 . . . . 5  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
13 eqid 2443 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
14 homfeq.j . . . . 5  |-  J  =  ( Hom  `  D
)
1512, 13, 14homffval 14635 . . . 4  |-  ( Hom f  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) )
1611, 15syl6reqr 2494 . . 3  |-  ( ph  ->  ( Hom f  `  D )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) ) )
178, 16eqeq12d 2457 . 2  |-  ( ph  ->  ( ( Hom f  `  C )  =  ( Hom f  `  D )  <-> 
( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  B , 
y  e.  B  |->  ( x J y ) ) ) )
18 ovex 6121 . . . 4  |-  ( x H y )  e. 
_V
1918rgen2w 2789 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x H y )  e. 
_V
20 mpt22eqb 6204 . . 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 261 1  |-  ( ph  ->  ( ( Hom f  `  C )  =  ( Hom f  `  D )  <->  A. x  e.  B  A. y  e.  B  ( x H y )  =  ( x J y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Basecbs 14179   Hom chom 14254   Hom f chomf 14609
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-homf 14613
This theorem is referenced by:  homfeqd  14639  fullresc  14766  resssetc  14965  resscatc  14978
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