Theorem xpchomfval 14981

Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
xpchomfval.t	|- T = ( C X.c D )
xpchomfval.y	|- B = ( Base ` T )
xpchomfval.h	|- H = ( Hom ` C )
xpchomfval.j	|- J = ( Hom ` D )
xpchomfval.k	|- K = ( Hom ` T )

Assertion

Ref	Expression
xpchomfval	|- K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )

Distinct variable groups:   v, u, B   u, C, v   u, D, v   u, H, v   u, J, v

Allowed substitution hints:   T( v, u)   K( v, u)

Proof of Theorem xpchomfval

Dummy variables f g x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpchomfval.t	. . . 4 |- T = ( C X.c D )
2		eqid 2438	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2438	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		xpchomfval.h	. . . 4 |- H = ( Hom ` C )
5		xpchomfval.j	. . . 4 |- J = ( Hom ` D )
6		eqid 2438	. . . 4 |- (comp ` C ) = (comp ` C )
7		eqid 2438	. . . 4 |- (comp ` D ) = (comp ` D )
8		simpl 457	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> C e. _V )
9		simpr 461	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> D e. _V )
10		xpchomfval.y	. . . . . 6 |- B = ( Base ` T )
11	1, 2, 3	xpcbas 14980	. . . . . 6 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
12	10, 11	eqtr4i 2461	. . . . 5 |- B = ( ( Base ` C ) X. ( Base ` D ) )
13	12	a1i 11	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> B = ( ( Base ` C ) X. ( Base ` D ) ) )
14		eqidd 2439	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
15		eqidd 2439	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) y ) , f e. ( ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` C ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) y ) , f e. ( ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` C ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15	xpcval 14979	. . 3 |- ( ( C e. _V /\ D e. _V ) -> T = { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) >. , <. (comp ` ndx ) , ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) y ) , f e. ( ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` C ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
17		catstr 14859	. . 3 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) >. , <. (comp ` ndx ) , ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) y ) , f e. ( ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` C ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } Struct <. 1 , ; 1 5 >.
18		homid 14346	. . 3 |- Hom = Slot ( Hom ` ndx )
19		snsstp2 4020	. . 3 |- { <. ( Hom ` ndx ) , ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) >. } C_ { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) >. , <. (comp ` ndx ) , ( x e. ( B X. B ) , y e. B |-> ( g e. ( ( 2nd ` x ) ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) y ) , f e. ( ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` C ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` D ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. }
20		fvex 5696	. . . . . 6 |- ( Base ` T ) e. _V
21	10, 20	eqeltri 2508	. . . . 5 |- B e. _V
22	21, 21	mpt2ex 6645	. . . 4 |- ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) e. _V
23	22	a1i 11	. . 3 |- ( ( C e. _V /\ D e. _V ) -> ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) e. _V )
24		xpchomfval.k	. . 3 |- K = ( Hom ` T )
25	16, 17, 18, 19, 23, 24	strfv3 14201	. 2 |- ( ( C e. _V /\ D e. _V ) -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
26		mpt20 6151	. . . 4 |- ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = (/)
27	26	eqcomi 2442	. . 3 |- (/) = ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
28		fnxpc 14978	. . . . . . . 8 |- X.c Fn ( _V X. _V )
29		fndm 5505	. . . . . . . 8 |- ( X.c Fn ( _V X. _V ) -> dom X.c = ( _V X. _V ) )
30	28, 29	ax-mp 5	. . . . . . 7 |- dom X.c = ( _V X. _V )
31	30	ndmov 6242	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> ( C X.c D ) = (/) )
32	1, 31	syl5eq 2482	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> T = (/) )
33	32	fveq2d 5690	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( Hom ` T ) = ( Hom ` (/) ) )
34	18	str0 14204	. . . 4 |- (/) = ( Hom ` (/) )
35	33, 24, 34	3eqtr4g 2495	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> K = (/) )
36	32	fveq2d 5690	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> ( Base ` T ) = ( Base ` (/) ) )
37		base0 14205	. . . . 5 |- (/) = ( Base ` (/) )
38	36, 10, 37	3eqtr4g 2495	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> B = (/) )
39		eqidd 2439	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) = ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
40	38, 38, 39	mpt2eq123dv 6143	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
41	27, 35, 40	3eqtr4a 2496	. 2 |- ( -. ( C e. _V /\ D e. _V ) -> K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) )
42	25, 41	pm2.61i 164	1 |- K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )

Syntax hints:   -. wn 3   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2967   (/)c0 3632   {ctp 3876   <.cop 3878   X. cxp 4833   dom cdm 4835   Fn wfn 5408   ` cfv 5413  (class class class)co 6086   e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571   1c1 9275   5c5 10366  ;cdc 10747   ndxcnx 14163   Basecbs 14166   Hom chom 14241  compcco 14242   X._c cxpc 14970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-hom 14254  df-cco 14255  df-xpc 14974

This theorem is referenced by:  xpchom  14982  relxpchom  14983  xpccofval  14984  catcxpccl  15009  xpcpropd  15010