Theorem xpchomfval 15111

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 2454	. . . 4 |- ( Base ` C ) = ( Base ` C )
3		eqid 2454	. . . 4 |- ( Base ` D ) = ( Base ` D )
4		xpchomfval.h	. . . 4 |- H = ( Hom ` C )
5		xpchomfval.j	. . . 4 |- J = ( Hom ` D )
6		eqid 2454	. . . 4 |- (comp ` C ) = (comp ` C )
7		eqid 2454	. . . 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 15110	. . . . . 6 |- ( ( Base ` C ) X. ( Base ` D ) ) = ( Base ` T )
12	10, 11	eqtr4i 2486	. . . . 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 2455	. . . 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 2455	. . . 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 15109	. . 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 14989	. . 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 14476	. . 3 |- Hom = Slot ( Hom ` ndx )
19		snsstp2 4136	. . 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 5812	. . . . . 6 |- ( Base ` T ) e. _V
21	10, 20	eqeltri 2538	. . . . 5 |- B e. _V
22	21, 21	mpt2ex 6763	. . . 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 14330	. 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 6268	. . . 4 |- ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) ) = (/)
27	26	eqcomi 2467	. . 3 |- (/) = ( u e. (/) , v e. (/) |-> ( ( ( 1st ` u ) H ( 1st ` v ) ) X. ( ( 2nd ` u ) J ( 2nd ` v ) ) ) )
28		fnxpc 15108	. . . . . . . 8 |- X.c Fn ( _V X. _V )
29		fndm 5621	. . . . . . . 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 6360	. . . . . 6 |- ( -. ( C e. _V /\ D e. _V ) -> ( C X.c D ) = (/) )
32	1, 31	syl5eq 2507	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> T = (/) )
33	32	fveq2d 5806	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> ( Hom ` T ) = ( Hom ` (/) ) )
34	18	str0 14333	. . . 4 |- (/) = ( Hom ` (/) )
35	33, 24, 34	3eqtr4g 2520	. . 3 |- ( -. ( C e. _V /\ D e. _V ) -> K = (/) )
36	32	fveq2d 5806	. . . . 5 |- ( -. ( C e. _V /\ D e. _V ) -> ( Base ` T ) = ( Base ` (/) ) )
37		base0 14334	. . . . 5 |- (/) = ( Base ` (/) )
38	36, 10, 37	3eqtr4g 2520	. . . 4 |- ( -. ( C e. _V /\ D e. _V ) -> B = (/) )
39		eqidd 2455	. . . 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 6260	. . 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 2521	. 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 1370   e. wcel 1758   _Vcvv 3078   (/)c0 3748   {ctp 3992   <.cop 3994   X. cxp 4949   dom cdm 4951   Fn wfn 5524   ` cfv 5529  (class class class)co 6203   |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   1c1 9397   5c5 10488  ;cdc 10869   ndxcnx 14292   Basecbs 14295   Hom chom 14371  compcco 14372   X._c cxpc 15100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-hom 14384  df-cco 14385  df-xpc 15104

This theorem is referenced by:  xpchom  15112  relxpchom  15113  xpccofval  15114  catcxpccl  15139  xpcpropd  15140