Theorem diagval 16880

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.

Step	Hyp	Ref	Expression
1		diagval.l	. 2 |- L = ( CΔfunc D )
2		df-diag 16856	. . . 4 |- Δfunc = ( c e. Cat , d e. Cat |-> ( <. c , d >. curryF ( c 1stF d ) ) )
3	2	a1i 11	. . 3 |- ( ph -> Δfunc = ( c e. Cat , d e. Cat |-> ( <. c , d >. curryF ( c 1stF d ) ) ) )
4		simprl 794	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> c = C )
5		simprr 796	. . . . 5 |- ( ( ph /\ ( c = C /\ d = D ) ) -> d = D )
6	4, 5	opeq12d 4410	. . . 4 |- ( ( ph /\ ( c = C /\ d = D ) ) -> <. c , d >. = <. C , D >. )
7	4, 5	oveq12d 6668	. . . 4 |- ( ( ph /\ ( c = C /\ d = D ) ) -> ( c 1stF d ) = ( C 1stF D ) )
8	6, 7	oveq12d 6668	. . 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		ovexd 6680	. . 3 |- ( ph -> ( <. C , D >. curryF ( C 1stF D ) ) e. _V )
12	3, 8, 9, 10, 11	ovmpt2d 6788	. 2 |- ( ph -> ( CΔfunc D ) = ( <. C , D >. curryF ( C 1stF D ) ) )
13	1, 12	syl5eq 2668	1 |- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183  (class class class)co 6650   |-> cmpt2 6652   Catccat 16325   1st_F c1stf 16809   curry_F ccurf 16850  Δ_funccdiag 16852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-diag 16856

This theorem is referenced by:  diagcl  16881  diag11  16883  diag12  16884  diag2  16885