Theorem diag2 16885

Description: Value of the diagonal functor at a morphism. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypotheses

Ref	Expression
diag2.l	|- L = ( CΔfunc D )
diag2.a	|- A = ( Base ` C )
diag2.b	|- B = ( Base ` D )
diag2.h	|- H = ( Hom ` C )
diag2.c	|- ( ph -> C e. Cat )
diag2.d	|- ( ph -> D e. Cat )
diag2.x	|- ( ph -> X e. A )
diag2.y	|- ( ph -> Y e. A )
diag2.f	|- ( ph -> F e. ( X H Y ) )

Assertion

Ref	Expression
diag2	|- ( ph -> ( ( X ( 2nd ` L ) Y ) ` F ) = ( B X. { F } ) )

Proof of Theorem diag2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diag2.l	. . . . . 6 |- L = ( CΔfunc D )
2		diag2.c	. . . . . 6 |- ( ph -> C e. Cat )
3		diag2.d	. . . . . 6 |- ( ph -> D e. Cat )
4	1, 2, 3	diagval 16880	. . . . 5 |- ( ph -> L = ( <. C , D >. curryF ( C 1stF D ) ) )
5	4	fveq2d 6195	. . . 4 |- ( ph -> ( 2nd ` L ) = ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) )
6	5	oveqd 6667	. . 3 |- ( ph -> ( X ( 2nd ` L ) Y ) = ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) )
7	6	fveq1d 6193	. 2 |- ( ph -> ( ( X ( 2nd ` L ) Y ) ` F ) = ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) )
8		eqid 2622	. . 3 |- ( <. C , D >. curryF ( C 1stF D ) ) = ( <. C , D >. curryF ( C 1stF D ) )
9		diag2.a	. . 3 |- A = ( Base ` C )
10		eqid 2622	. . . 4 |- ( C X.c D ) = ( C X.c D )
11		eqid 2622	. . . 4 |- ( C 1stF D ) = ( C 1stF D )
12	10, 2, 3, 11	1stfcl 16837	. . 3 |- ( ph -> ( C 1stF D ) e. ( ( C X.c D ) Func C ) )
13		diag2.b	. . 3 |- B = ( Base ` D )
14		diag2.h	. . 3 |- H = ( Hom ` C )
15		eqid 2622	. . 3 |- ( Id ` D ) = ( Id ` D )
16		diag2.x	. . 3 |- ( ph -> X e. A )
17		diag2.y	. . 3 |- ( ph -> Y e. A )
18		diag2.f	. . 3 |- ( ph -> F e. ( X H Y ) )
19		eqid 2622	. . 3 |- ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) = ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F )
20	8, 9, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19	curf2 16869	. 2 |- ( ph -> ( ( X ( 2nd ` ( <. C , D >. curryF ( C 1stF D ) ) ) Y ) ` F ) = ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) )
21	10, 9, 13	xpcbas 16818	. . . . . . 7 |- ( A X. B ) = ( Base ` ( C X.c D ) )
22		eqid 2622	. . . . . . 7 |- ( Hom ` ( C X.c D ) ) = ( Hom ` ( C X.c D ) )
23	2	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. B ) -> C e. Cat )
24	3	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. B ) -> D e. Cat )
25		opelxpi 5148	. . . . . . . 8 |- ( ( X e. A /\ x e. B ) -> <. X , x >. e. ( A X. B ) )
26	16, 25	sylan 488	. . . . . . 7 |- ( ( ph /\ x e. B ) -> <. X , x >. e. ( A X. B ) )
27		opelxpi 5148	. . . . . . . 8 |- ( ( Y e. A /\ x e. B ) -> <. Y , x >. e. ( A X. B ) )
28	17, 27	sylan 488	. . . . . . 7 |- ( ( ph /\ x e. B ) -> <. Y , x >. e. ( A X. B ) )
29	10, 21, 22, 23, 24, 11, 26, 28	1stf2 16833	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) = ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) )
30	29	oveqd 6667	. . . . 5 |- ( ( ph /\ x e. B ) -> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) = ( F ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) )
31		df-ov 6653	. . . . . 6 |- ( F ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) = ( ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ` <. F , ( ( Id ` D ) ` x ) >. )
32	18	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> F e. ( X H Y ) )
33		eqid 2622	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
34		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. B ) -> x e. B )
35	13, 33, 15, 24, 34	catidcl 16343	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> ( ( Id ` D ) ` x ) e. ( x ( Hom ` D ) x ) )
36		opelxpi 5148	. . . . . . . . 9 |- ( ( F e. ( X H Y ) /\ ( ( Id ` D ) ` x ) e. ( x ( Hom ` D ) x ) ) -> <. F , ( ( Id ` D ) ` x ) >. e. ( ( X H Y ) X. ( x ( Hom ` D ) x ) ) )
37	32, 35, 36	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. B ) -> <. F , ( ( Id ` D ) ` x ) >. e. ( ( X H Y ) X. ( x ( Hom ` D ) x ) ) )
38	16	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> X e. A )
39	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. B ) -> Y e. A )
40	10, 9, 13, 14, 33, 38, 34, 39, 34, 22	xpchom2 16826	. . . . . . . 8 |- ( ( ph /\ x e. B ) -> ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) = ( ( X H Y ) X. ( x ( Hom ` D ) x ) ) )
41	37, 40	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ x e. B ) -> <. F , ( ( Id ` D ) ` x ) >. e. ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) )
42		fvres 6207	. . . . . . 7 |- ( <. F , ( ( Id ` D ) ` x ) >. e. ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) -> ( ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ` <. F , ( ( Id ` D ) ` x ) >. ) = ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) )
43	41, 42	syl 17	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ` <. F , ( ( Id ` D ) ` x ) >. ) = ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) )
44	31, 43	syl5eq 2668	. . . . 5 |- ( ( ph /\ x e. B ) -> ( F ( 1st |` ( <. X , x >. ( Hom ` ( C X.c D ) ) <. Y , x >. ) ) ( ( Id ` D ) ` x ) ) = ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) )
45		op1stg 7180	. . . . . 6 |- ( ( F e. ( X H Y ) /\ ( ( Id ` D ) ` x ) e. ( x ( Hom ` D ) x ) ) -> ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) = F )
46	32, 35, 45	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. B ) -> ( 1st ` <. F , ( ( Id ` D ) ` x ) >. ) = F )
47	30, 44, 46	3eqtrd 2660	. . . 4 |- ( ( ph /\ x e. B ) -> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) = F )
48	47	mpteq2dva 4744	. . 3 |- ( ph -> ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) = ( x e. B |-> F ) )
49		fconstmpt 5163	. . 3 |- ( B X. { F } ) = ( x e. B |-> F )
50	48, 49	syl6eqr 2674	. 2 |- ( ph -> ( x e. B |-> ( F ( <. X , x >. ( 2nd ` ( C 1stF D ) ) <. Y , x >. ) ( ( Id ` D ) ` x ) ) ) = ( B X. { F } ) )
51	7, 20, 50	3eqtrd 2660	1 |- ( ph -> ( ( X ( 2nd ` L ) Y ) ` F ) = ( B X. { F } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Catccat 16325   Idccid 16326   X._c cxpc 16808   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-xpc 16812  df-1stf 16813  df-curf 16854  df-diag 16856

This theorem is referenced by:  diag2cl  16886