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Theorem homfeq 15110
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 2458 . . . . 5  |-  ( ph  ->  ( x H y )  =  ( x H y ) )
31, 1, 2mpt2eq123dv 6358 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x H y ) )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) ) )
4 eqid 2457 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
5 eqid 2457 . . . . 5  |-  ( Base `  C )  =  (
Base `  C )
6 homfeq.h . . . . 5  |-  H  =  ( Hom  `  C
)
74, 5, 6homffval 15106 . . . 4  |-  ( Hom f  `  C )  =  ( x  e.  ( Base `  C ) ,  y  e.  ( Base `  C
)  |->  ( x H y ) )
83, 7syl6reqr 2517 . . 3  |-  ( ph  ->  ( Hom f  `  C )  =  ( x  e.  B ,  y  e.  B  |->  ( x H y ) ) )
9 homfeq.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  D ) )
10 eqidd 2458 . . . . 5  |-  ( ph  ->  ( x J y )  =  ( x J y ) )
119, 9, 10mpt2eq123dv 6358 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  ( x J y ) )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) ) )
12 eqid 2457 . . . . 5  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
13 eqid 2457 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
14 homfeq.j . . . . 5  |-  J  =  ( Hom  `  D
)
1512, 13, 14homffval 15106 . . . 4  |-  ( Hom f  `  D )  =  ( x  e.  ( Base `  D ) ,  y  e.  ( Base `  D
)  |->  ( x J y ) )
1611, 15syl6reqr 2517 . . 3  |-  ( ph  ->  ( Hom f  `  D )  =  ( x  e.  B ,  y  e.  B  |->  ( x J y ) ) )
178, 16eqeq12d 2479 . 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 6324 . . . 4  |-  ( x H y )  e. 
_V
1918rgen2w 2819 . . 3  |-  A. x  e.  B  A. y  e.  B  ( x H y )  e. 
_V
20 mpt22eqb 6410 . . 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 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Basecbs 14644   Hom chom 14723   Hom f chomf 15083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-homf 15087
This theorem is referenced by:  homfeqd  15111  fullresc  15267  resssetc  15498  resscatc  15511
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