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Theorem 2ndfval 14254
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 5709 . . . . . . 7  |-  ( Base `  c )  e.  _V
5 fvex 5709 . . . . . . 7  |-  ( Base `  d )  e.  _V
64, 5xpex 4957 . . . . . 6  |-  ( (
Base `  c )  X.  ( Base `  d
) )  e.  _V
76a1i 11 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  e. 
_V )
8 simpl 444 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  c  =  C )
98fveq2d 5699 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 simpr 448 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1110fveq2d 5699 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  d
)  =  ( Base `  D ) )
129, 11xpeq12d 4870 . . . . . 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 2412 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2412 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
1613, 14, 15xpcbas 14238 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
17 1stfval.b . . . . . . 7  |-  B  =  ( Base `  T
)
1816, 17eqtr4i 2435 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  B
1912, 18syl6eq 2460 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  B )
20 simpr 448 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2120reseq2d 5113 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  b )  =  ( 2nd  |`  B ) )
22 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  c  =  C )
23 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
2422, 23oveq12d 6066 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  ( C  X.c  D ) )
2524, 13syl6eqr 2462 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  T )
2625fveq2d 5699 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  (  Hom  `  T )
)
27 1stfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  T
)
2826, 27syl6eqr 2462 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  H )
2928oveqd 6065 . . . . . . . 8  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x (  Hom  `  (
c  X.c  d ) ) y )  =  ( x H y ) )
3029reseq2d 5113 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 2nd  |`  ( x (  Hom  `  ( c  X.c  d ) ) y ) )  =  ( 2nd  |`  ( x H y ) ) )
3120, 20, 30mpt2eq123dv 6103 . . . . . 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 3960 . . . . 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 3262 . . . 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 14234 . . . 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 4395 . . . 4  |-  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  e.  _V
3633, 34, 35ovmpt2a 6171 . . 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 643 . 2  |-  ( ph  ->  ( C  2ndF  D )  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
381, 37syl5eq 2456 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2924   [_csb 3219   <.cop 3785    X. cxp 4843    |` cres 4847   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   2ndc2nd 6315   Basecbs 13432    Hom chom 13503   Catccat 13852    X.c cxpc 14228    2ndF c2ndf 14230
This theorem is referenced by:  2ndf1  14255  2ndf2  14256  2ndfcl  14258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-hom 13516  df-cco 13517  df-xpc 14232  df-2ndf 14234
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