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Theorem xpchom 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
)
xpchom.x  |-  ( ph  ->  X  e.  B )
xpchom.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
xpchom  |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )

Proof of Theorem xpchom
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchom.x . 2  |-  ( ph  ->  X  e.  B )
2 xpchom.y . 2  |-  ( ph  ->  Y  e.  B )
3 simpl 457 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  u  =  X )
43fveq2d 5876 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  u
)  =  ( 1st `  X ) )
5 simpr 461 . . . . . 6  |-  ( ( u  =  X  /\  v  =  Y )  ->  v  =  Y )
65fveq2d 5876 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 1st `  v
)  =  ( 1st `  Y ) )
74, 6oveq12d 6313 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 1st `  u
) H ( 1st `  v ) )  =  ( ( 1st `  X
) H ( 1st `  Y ) ) )
83fveq2d 5876 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  u
)  =  ( 2nd `  X ) )
95fveq2d 5876 . . . . 5  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( 2nd `  v
)  =  ( 2nd `  Y ) )
108, 9oveq12d 6313 . . . 4  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( 2nd `  u
) J ( 2nd `  v ) )  =  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )
117, 10xpeq12d 5030 . . 3  |-  ( ( u  =  X  /\  v  =  Y )  ->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) )  =  ( ( ( 1st `  X ) H ( 1st `  Y ) )  X.  ( ( 2nd `  X ) J ( 2nd `  Y
) ) ) )
12 xpchomfval.t . . . 4  |-  T  =  ( C  X.c  D )
13 xpchomfval.y . . . 4  |-  B  =  ( Base `  T
)
14 xpchomfval.h . . . 4  |-  H  =  ( Hom  `  C
)
15 xpchomfval.j . . . 4  |-  J  =  ( Hom  `  D
)
16 xpchomfval.k . . . 4  |-  K  =  ( Hom  `  T
)
1712, 13, 14, 15, 16xpchomfval 15322 . . 3  |-  K  =  ( u  e.  B ,  v  e.  B  |->  ( ( ( 1st `  u ) H ( 1st `  v ) )  X.  ( ( 2nd `  u ) J ( 2nd `  v
) ) ) )
18 ovex 6320 . . . 4  |-  ( ( 1st `  X ) H ( 1st `  Y
) )  e.  _V
19 ovex 6320 . . . 4  |-  ( ( 2nd `  X ) J ( 2nd `  Y
) )  e.  _V
2018, 19xpex 6599 . . 3  |-  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) )  e.  _V
2111, 17, 20ovmpt2a 6428 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
221, 2, 21syl2anc 661 1  |-  ( ph  ->  ( X K Y )  =  ( ( ( 1st `  X
) H ( 1st `  Y ) )  X.  ( ( 2nd `  X
) J ( 2nd `  Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   Basecbs 14506   Hom chom 14582    X.c cxpc 15311
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 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-hom 14595  df-cco 14596  df-xpc 15315
This theorem is referenced by:  xpchom2  15329  xpccatid  15331  1stfcl  15340  2ndfcl  15341  xpcpropd  15351  evlfcl  15365  curf1cl  15371  hofcl  15402  yonedalem3  15423
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