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Theorem diagval 15626
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 15602 . . . 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 754 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 755 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5opeq12d 4139 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. c ,  d >.  =  <. C ,  D >. )
74, 5oveq12d 6214 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  1stF  d )  =  ( C  1stF  D ) )
86, 7oveq12d 6214 . . 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 6224 . . . 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 6329 . 2  |-  ( ph  ->  ( CΔfunc D )  =  (
<. C ,  D >. curryF  ( C  1stF  D ) ) )
141, 13syl5eq 2435 1  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   <.cop 3950  (class class class)co 6196    |-> cmpt2 6198   Catccat 15071    1stF c1stf 15555   curryF ccurf 15596  Δfunccdiag 15598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-diag 15602
This theorem is referenced by:  diagcl  15627  diag11  15629  diag12  15630  diag2  15631
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