Theorem xpchomfval 15323

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 2467	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2467	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		xpchomfval.h	. . . 4 |- H = ( Hom ` C )
5		xpchomfval.j	. . . 4 |- J = ( Hom ` D )
6		eqid 2467	. . . 4 |- (comp ` C ) = (comp ` C )
7		eqid 2467	. . . 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 15322	. . . . . 6 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
12	10, 11	eqtr4i 2499	. . . . 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 2468	. . . 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 2468	. . . 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 15321	. . 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 15201	. . 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 14688	. . 3 |- Hom = Slot ( Hom ` ndx )
19		snsstp2 4185	. . 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 5882	. . . . . 6 |- ( Base ` T ) e. _V
21	10, 20	eqeltri 2551	. . . . 5 |- B e. _V
22	21, 21	mpt2ex 6872	. . . 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 14542	. 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 6362	. . . 4 |- ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = (/)
27	26	eqcomi 2480	. . 3 |- (/) = ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
28		fnxpc 15320	. . . . . . . 8 |- X.c Fn ( _V X. _V )
29		fndm 5686	. . . . . . . 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 6454	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> ( C X.c D ) = (/) )
32	1, 31	syl5eq 2520	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> T = (/) )
33	32	fveq2d 5876	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( Hom ` T ) = ( Hom ` (/) ) )
34	18	str0 14545	. . . 4 |- (/) = ( Hom ` (/) )
35	33, 24, 34	3eqtr4g 2533	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> K = (/) )
36	32	fveq2d 5876	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> ( Base ` T ) = ( Base ` (/) ) )
37		base0 14546	. . . . 5 |- (/) = ( Base ` (/) )
38	36, 10, 37	3eqtr4g 2533	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> B = (/) )
39		eqidd 2468	. . . 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 6354	. . 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 2534	. 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 1379   e. wcel 1767   _Vcvv 3118   (/)c0 3790   {ctp 4037   <.cop 4039   X. cxp 5003   dom cdm 5005   Fn wfn 5589   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   1c1 9505   5c5 10600  ;cdc 10988   ndxcnx 14504   Basecbs 14507   Hom chom 14583  compcco 14584   X._c cxpc 15312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-xpc 15316

This theorem is referenced by:  xpchom  15324  relxpchom  15325  xpccofval  15326  catcxpccl  15351  xpcpropd  15352