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Theorem diagval 14030
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 14006 . . . 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 3820 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. c ,  d >.  =  <. C ,  D >. )
74, 5oveq12d 5892 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  1stF  d )  =  ( C  1stF  D ) )
86, 7oveq12d 5892 . . 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 5899 . . . 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 5991 . 2  |-  ( ph  ->  ( CΔfunc D )  =  (
<. C ,  D >. curryF  ( C  1stF  D ) ) )
141, 13syl5eq 2340 1  |-  ( ph  ->  L  =  ( <. C ,  D >. curryF  ( C  1stF  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656  (class class class)co 5874    e. cmpt2 5876   Catccat 13582    1stF c1stf 13959   curryF ccurf 14000  Δfunccdiag 14002
This theorem is referenced by:  diagcl  14031  diag11  14033  diag12  14034  diag2  14035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-diag 14006
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