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Theorem 2ndfval 14996
Description: Value of the first projection functor. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfval.t  |-  T  =  ( C  X.c  D )
1stfval.b  |-  B  =  ( Base `  T
)
1stfval.h  |-  H  =  ( Hom  `  T
)
1stfval.c  |-  ( ph  ->  C  e.  Cat )
1stfval.d  |-  ( ph  ->  D  e.  Cat )
2ndfval.p  |-  Q  =  ( C  2ndF  D )
Assertion
Ref Expression
2ndfval  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
Distinct variable groups:    x, y, B    x, C, y    x, D, y    x, H, y    ph, x, y
Allowed substitution hints:    Q( x, y)    T( x, y)

Proof of Theorem 2ndfval
Dummy variables  b 
c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2ndfval.p . 2  |-  Q  =  ( C  2ndF  D )
2 1stfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 1stfval.d . . 3  |-  ( ph  ->  D  e.  Cat )
4 fvex 5696 . . . . . . 7  |-  ( Base `  c )  e.  _V
5 fvex 5696 . . . . . . 7  |-  ( Base `  d )  e.  _V
64, 5xpex 6503 . . . . . 6  |-  ( (
Base `  c )  X.  ( Base `  d
) )  e.  _V
76a1i 11 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  e. 
_V )
8 simpl 457 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  c  =  C )
98fveq2d 5690 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 simpr 461 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1110fveq2d 5690 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  d
)  =  ( Base `  D ) )
129, 11xpeq12d 4860 . . . . . 6  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  ( ( Base `  C
)  X.  ( Base `  D ) ) )
13 1stfval.t . . . . . . . 8  |-  T  =  ( C  X.c  D )
14 eqid 2438 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2438 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
1613, 14, 15xpcbas 14980 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
17 1stfval.b . . . . . . 7  |-  B  =  ( Base `  T
)
1816, 17eqtr4i 2461 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  B
1912, 18syl6eq 2486 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  B )
20 simpr 461 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2120reseq2d 5105 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  b )  =  ( 2nd  |`  B ) )
22 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  c  =  C )
23 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
2422, 23oveq12d 6104 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  ( C  X.c  D ) )
2524, 13syl6eqr 2488 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  T )
2625fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( Hom  `  ( c  X.c  d ) )  =  ( Hom  `  T )
)
27 1stfval.h . . . . . . . . . 10  |-  H  =  ( Hom  `  T
)
2826, 27syl6eqr 2488 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( Hom  `  ( c  X.c  d ) )  =  H )
2928oveqd 6103 . . . . . . . 8  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x ( Hom  `  (
c  X.c  d ) ) y )  =  ( x H y ) )
3029reseq2d 5105 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  ( x ( Hom  `  ( c  X.c  d ) ) y ) )  =  ( 2nd  |`  ( x H y ) ) )
3120, 20, 30mpt2eq123dv 6143 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( 2nd  |`  ( x
( Hom  `  ( c  X.c  d ) ) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) )
3221, 31opeq12d 4062 . . . . 5  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  <. ( 2nd  |`  b ) ,  ( x  e.  b ,  y  e.  b 
|->  ( 2nd  |`  (
x ( Hom  `  (
c  X.c  d ) ) y ) ) ) >.  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
337, 19, 32csbied2 3310 . . . 4  |-  ( ( c  =  C  /\  d  =  D )  ->  [_ ( ( Base `  c )  X.  ( Base `  d ) )  /  b ]_ <. ( 2nd  |`  b ) ,  ( x  e.  b ,  y  e.  b  |->  ( 2nd  |`  (
x ( Hom  `  (
c  X.c  d ) ) y ) ) ) >.  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
34 df-2ndf 14976 . . . 4  |-  2ndF  =  (
c  e.  Cat , 
d  e.  Cat  |->  [_ ( ( Base `  c
)  X.  ( Base `  d ) )  / 
b ]_ <. ( 2nd  |`  b
) ,  ( x  e.  b ,  y  e.  b  |->  ( 2nd  |`  ( x ( Hom  `  ( c  X.c  d ) ) y ) ) ) >. )
35 opex 4551 . . . 4  |-  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  e.  _V
3633, 34, 35ovmpt2a 6216 . . 3  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C  2ndF  D )  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
372, 3, 36syl2anc 661 . 2  |-  ( ph  ->  ( C  2ndF  D )  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
381, 37syl5eq 2482 1  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   [_csb 3283   <.cop 3878    X. cxp 4833    |` cres 4837   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   2ndc2nd 6571   Basecbs 14166   Hom chom 14241   Catccat 14594    X.c cxpc 14970    2ndF c2ndf 14972
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  df-2ndf 14976
This theorem is referenced by:  2ndf1  14997  2ndf2  14998  2ndfcl  15000
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