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Theorem fthpropd 14813
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fullpropd.1  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
fullpropd.2  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
fullpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
fullpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
fullpropd.a  |-  ( ph  ->  A  e.  V )
fullpropd.b  |-  ( ph  ->  B  e.  V )
fullpropd.c  |-  ( ph  ->  C  e.  V )
fullpropd.d  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
fthpropd  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )

Proof of Theorem fthpropd
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfth 14801 . 2  |-  Rel  ( A Faith  C )
2 relfth 14801 . 2  |-  Rel  ( B Faith  D )
3 fullpropd.1 . . . . . 6  |-  ( ph  ->  ( Hom f  `  A )  =  ( Hom f  `  B ) )
4 fullpropd.2 . . . . . 6  |-  ( ph  ->  (compf `  A )  =  (compf `  B ) )
5 fullpropd.3 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
6 fullpropd.4 . . . . . 6  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
7 fullpropd.a . . . . . 6  |-  ( ph  ->  A  e.  V )
8 fullpropd.b . . . . . 6  |-  ( ph  ->  B  e.  V )
9 fullpropd.c . . . . . 6  |-  ( ph  ->  C  e.  V )
10 fullpropd.d . . . . . 6  |-  ( ph  ->  D  e.  V )
113, 4, 5, 6, 7, 8, 9, 10funcpropd 14792 . . . . 5  |-  ( ph  ->  ( A  Func  C
)  =  ( B 
Func  D ) )
1211breqd 4291 . . . 4  |-  ( ph  ->  ( f ( A 
Func  C ) g  <->  f ( B  Func  D ) g ) )
133homfeqbas 14617 . . . . 5  |-  ( ph  ->  ( Base `  A
)  =  ( Base `  B ) )
1413raleqdv 2913 . . . . 5  |-  ( ph  ->  ( A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1513, 14raleqbidv 2921 . . . 4  |-  ( ph  ->  ( A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y )  <->  A. x  e.  ( Base `  B ) A. y  e.  ( Base `  B ) Fun  `' ( x g y ) ) )
1612, 15anbi12d 703 . . 3  |-  ( ph  ->  ( ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y ) )  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) Fun  `' (
x g y ) ) ) )
17 eqid 2433 . . . 4  |-  ( Base `  A )  =  (
Base `  A )
1817isfth 14806 . . 3  |-  ( f ( A Faith  C ) g  <->  ( f ( A  Func  C )
g  /\  A. x  e.  ( Base `  A
) A. y  e.  ( Base `  A
) Fun  `' (
x g y ) ) )
19 eqid 2433 . . . 4  |-  ( Base `  B )  =  (
Base `  B )
2019isfth 14806 . . 3  |-  ( f ( B Faith  D ) g  <->  ( f ( B  Func  D )
g  /\  A. x  e.  ( Base `  B
) A. y  e.  ( Base `  B
) Fun  `' (
x g y ) ) )
2116, 18, 203bitr4g 288 . 2  |-  ( ph  ->  ( f ( A Faith 
C ) g  <->  f ( B Faith  D ) g ) )
221, 2, 21eqbrrdiv 4925 1  |-  ( ph  ->  ( A Faith  C )  =  ( B Faith  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   class class class wbr 4280   `'ccnv 4826   Fun wfun 5400   ` cfv 5406  (class class class)co 6080   Basecbs 14156   Hom f chomf 14586  compfccomf 14587    Func cfunc 14746   Faith cfth 14795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-map 7204  df-ixp 7252  df-cat 14588  df-cid 14589  df-homf 14590  df-comf 14591  df-func 14750  df-fth 14797
This theorem is referenced by: (None)
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