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Theorem diagval 14339
Description: Define the diagonal functor, which is the functor  C --> ( D  Func  C ) whose object part is  x  e.  C  |->  ( y  e.  D  |->  x ). We can define this equationally as the currying of the first projection functor, and by expressing it this way we get a quick proof of functoriality. (Contributed by Mario Carneiro, 6-Jan-2017.) (Revised by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
diagval.l  |-  L  =  ( CΔfunc D )
diagval.c  |-  ( ph  ->  C  e.  Cat )
diagval.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
diagval  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )

Proof of Theorem diagval
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diagval.l . 2  |-  L  =  ( CΔfunc D )
2 df-diag 14315 . . . 4  |- Δfunc  =  ( c  e.  Cat ,  d  e. 
Cat  |->  ( <. c ,  d >. curryF  ( c  1stF  d )
) )
32a1i 11 . . 3  |-  ( ph  -> Δfunc  =  ( c  e.  Cat ,  d  e.  Cat  |->  (
<. c ,  d >. curryF  ( c  1stF  d ) ) ) )
4 simprl 734 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 735 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5opeq12d 3994 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. c ,  d >.  =  <. C ,  D >. )
74, 5oveq12d 6101 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  1stF  d )  =  ( C  1stF  D ) )
86, 7oveq12d 6101 . . 3  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( <. c ,  d
>. curryF  ( c  1stF  d ) )  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
9 diagval.c . . 3  |-  ( ph  ->  C  e.  Cat )
10 diagval.d . . 3  |-  ( ph  ->  D  e.  Cat )
11 ovex 6108 . . . 4  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  e. 
_V
1211a1i 11 . . 3  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  1stF  D ) )  e. 
_V )
133, 8, 9, 10, 12ovmpt2d 6203 . 2  |-  ( ph  ->  ( CΔfunc D )  =  (
<. C ,  D >. curryF  ( C  1stF  D ) ) )
141, 13syl5eq 2482 1  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819  (class class class)co 6083    e. cmpt2 6085   Catccat 13891    1stF c1stf 14268   curryF ccurf 14309  Δfunccdiag 14311
This theorem is referenced by:  diagcl  14340  diag11  14342  diag12  14343  diag2  14344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-diag 14315
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