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Theorem diagval 14014
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 13990 . . . 4  |- Δfunc  =  ( c  e.  Cat ,  d  e. 
Cat  |->  ( <. c ,  d >. curryF  ( c  1stF  d )
) )
32a1i 10 . . 3  |-  ( ph  -> Δfunc  =  ( c  e.  Cat ,  d  e.  Cat  |->  (
<. c ,  d >. curryF  ( c  1stF  d ) ) ) )
4 simprl 732 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 733 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5opeq12d 3804 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. c ,  d >.  =  <. C ,  D >. )
74, 5oveq12d 5876 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  1stF  d )  =  ( C  1stF  D ) )
86, 7oveq12d 5876 . . 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 5883 . . . 4  |-  ( <. C ,  D >. curryF  ( C  1stF  D ) )  e. 
_V
1211a1i 10 . . 3  |-  ( ph  ->  ( <. C ,  D >. curryF  ( C  1stF  D ) )  e. 
_V )
133, 8, 9, 10, 12ovmpt2d 5975 . 2  |-  ( ph  ->  ( CΔfunc D )  =  (
<. C ,  D >. curryF  ( C  1stF  D ) ) )
141, 13syl5eq 2327 1  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643  (class class class)co 5858    e. cmpt2 5860   Catccat 13566    1stF c1stf 13943   curryF ccurf 13984  Δfunccdiag 13986
This theorem is referenced by:  diagcl  14015  diag11  14017  diag12  14018  diag2  14019
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-diag 13990
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