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Theorem 2ndfval 14292
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 5743 . . . . . . 7  |-  ( Base `  c )  e.  _V
5 fvex 5743 . . . . . . 7  |-  ( Base `  d )  e.  _V
64, 5xpex 4991 . . . . . 6  |-  ( (
Base `  c )  X.  ( Base `  d
) )  e.  _V
76a1i 11 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  e. 
_V )
8 simpl 445 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  c  =  C )
98fveq2d 5733 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 simpr 449 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1110fveq2d 5733 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  d
)  =  ( Base `  D ) )
129, 11xpeq12d 4904 . . . . . 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 2437 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2437 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
1613, 14, 15xpcbas 14276 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
17 1stfval.b . . . . . . 7  |-  B  =  ( Base `  T
)
1816, 17eqtr4i 2460 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  B
1912, 18syl6eq 2485 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  B )
20 simpr 449 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2120reseq2d 5147 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  b )  =  ( 2nd  |`  B ) )
22 simpll 732 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  c  =  C )
23 simplr 733 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
2422, 23oveq12d 6100 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  ( C  X.c  D ) )
2524, 13syl6eqr 2487 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  T )
2625fveq2d 5733 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  (  Hom  `  T )
)
27 1stfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  T
)
2826, 27syl6eqr 2487 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  H )
2928oveqd 6099 . . . . . . . 8  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x (  Hom  `  (
c  X.c  d ) ) y )  =  ( x H y ) )
3029reseq2d 5147 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  ( x (  Hom  `  ( c  X.c  d ) ) y ) )  =  ( 2nd  |`  ( x H y ) ) )
3120, 20, 30mpt2eq123dv 6137 . . . . . 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 3993 . . . . 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 3295 . . . 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 14272 . . . 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 4428 . . . 4  |-  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  e.  _V
3633, 34, 35ovmpt2a 6205 . . 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 644 . 2  |-  ( ph  ->  ( C  2ndF  D )  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
381, 37syl5eq 2481 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 360    = wceq 1653    e. wcel 1726   _Vcvv 2957   [_csb 3252   <.cop 3818    X. cxp 4877    |` cres 4881   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084   2ndc2nd 6349   Basecbs 13470    Hom chom 13541   Catccat 13890    X.c cxpc 14266    2ndF c2ndf 14268
This theorem is referenced by:  2ndf1  14293  2ndf2  14294  2ndfcl  14296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-hom 13554  df-cco 13555  df-xpc 14270  df-2ndf 14272
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