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Theorem diagval 15138
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 15114 . . . 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 755 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5opeq12d 4151 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. c ,  d >.  =  <. C ,  D >. )
74, 5oveq12d 6194 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  1stF  d )  =  ( C  1stF  D ) )
86, 7oveq12d 6194 . . 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 6201 . . . 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 6304 . 2  |-  ( ph  ->  ( CΔfunc D )  =  (
<. C ,  D >. curryF  ( C  1stF  D ) ) )
141, 13syl5eq 2502 1  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   _Vcvv 3054   <.cop 3967  (class class class)co 6176    |-> cmpt2 6178   Catccat 14690    1stF c1stf 15067   curryF ccurf 15108  Δfunccdiag 15110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-iota 5465  df-fun 5504  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-diag 15114
This theorem is referenced by:  diagcl  15139  diag11  15141  diag12  15142  diag2  15143
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