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Theorem 2ndfval 14178
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 5646 . . . . . . 7  |-  ( Base `  c )  e.  _V
5 fvex 5646 . . . . . . 7  |-  ( Base `  d )  e.  _V
64, 5xpex 4904 . . . . . 6  |-  ( (
Base `  c )  X.  ( Base `  d
) )  e.  _V
76a1i 10 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  e. 
_V )
8 simpl 443 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  c  =  C )
98fveq2d 5636 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 simpr 447 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1110fveq2d 5636 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  d
)  =  ( Base `  D ) )
129, 11xpeq12d 4817 . . . . . 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 2366 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2366 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
1613, 14, 15xpcbas 14162 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
17 1stfval.b . . . . . . 7  |-  B  =  ( Base `  T
)
1816, 17eqtr4i 2389 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  B
1912, 18syl6eq 2414 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  B )
20 simpr 447 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2120reseq2d 5058 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  b )  =  ( 2nd  |`  B ) )
22 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  c  =  C )
23 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
2422, 23oveq12d 5999 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  ( C  X.c  D ) )
2524, 13syl6eqr 2416 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  T )
2625fveq2d 5636 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  (  Hom  `  T )
)
27 1stfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  T
)
2826, 27syl6eqr 2416 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  H )
2928oveqd 5998 . . . . . . . 8  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x (  Hom  `  (
c  X.c  d ) ) y )  =  ( x H y ) )
3029reseq2d 5058 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  ( x (  Hom  `  ( c  X.c  d ) ) y ) )  =  ( 2nd  |`  ( x H y ) ) )
3120, 20, 30mpt2eq123dv 6036 . . . . . 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 3906 . . . . 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 3210 . . . 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 14158 . . . 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 4340 . . . 4  |-  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  e.  _V
3633, 34, 35ovmpt2a 6104 . . 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 642 . 2  |-  ( ph  ->  ( C  2ndF  D )  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
381, 37syl5eq 2410 1  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   [_csb 3167   <.cop 3732    X. cxp 4790    |` cres 4794   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   2ndc2nd 6248   Basecbs 13356    Hom chom 13427   Catccat 13776    X.c cxpc 14152    2ndF c2ndf 14154
This theorem is referenced by:  2ndf1  14179  2ndf2  14180  2ndfcl  14182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-hom 13440  df-cco 13441  df-xpc 14156  df-2ndf 14158
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