Theorem xpchomfval 16819

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 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2622	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		xpchomfval.h	. . . 4 |- H = ( Hom ` C )
5		xpchomfval.j	. . . 4 |- J = ( Hom ` D )
6		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
7		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
8		simpl 473	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> C e. _V )
9		simpr 477	. . . 4 |- ( ( C e. _V /\ D e. _V ) -> D e. _V )
10		xpchomfval.y	. . . . . 6 |- B = ( Base ` T )
11	1, 2, 3	xpcbas 16818	. . . . . 6 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
12	10, 11	eqtr4i 2647	. . . . 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 2623	. . . 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 2623	. . . 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 16817	. . 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 16617	. . 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 16075	. . 3 |- Hom = Slot ( Hom ` ndx )
19		snsstp2 4348	. . 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 6201	. . . . . 6 |- ( Base ` T ) e. _V
21	10, 20	eqeltri 2697	. . . . 5 |- B e. _V
22	21, 21	mpt2ex 7247	. . . 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 15908	. 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 6725	. . . 4 |- ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = (/)
27	26	eqcomi 2631	. . 3 |- (/) = ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
28		fnxpc 16816	. . . . . . . 8 |- X.c Fn ( _V X. _V )
29		fndm 5990	. . . . . . . 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 6818	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> ( C X.c D ) = (/) )
32	1, 31	syl5eq 2668	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> T = (/) )
33	32	fveq2d 6195	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( Hom ` T ) = ( Hom ` (/) ) )
34	18	str0 15911	. . . 4 |- (/) = ( Hom ` (/) )
35	33, 24, 34	3eqtr4g 2681	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> K = (/) )
36	32	fveq2d 6195	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> ( Base ` T ) = ( Base ` (/) ) )
37		base0 15912	. . . . 5 |- (/) = ( Base ` (/) )
38	36, 10, 37	3eqtr4g 2681	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> B = (/) )
39		eqidd 2623	. . . 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 6717	. . 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 2682	. 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 176	1 |- K = ( u e. B , v e. B |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {ctp 4181   <.cop 4183   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   1c1 9937   5c5 11073  ;cdc 11493   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   X._c cxpc 16808

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-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-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-xpc 16812

This theorem is referenced by:  xpchom  16820  relxpchom  16821  xpccofval  16822  catcxpccl  16847  xpcpropd  16848