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Theorem 1stfval 14288
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 )
1stfval.p  |-  P  =  ( C  1stF  D )
Assertion
Ref Expression
1stfval  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
Distinct variable groups:    x, y, B    x, C, y    x, D, y    x, H, y    ph, x, y
Allowed substitution hints:    P( x, y)    T( x, y)

Proof of Theorem 1stfval
Dummy variables  b 
c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.p . 2  |-  P  =  ( C  1stF  D )
2 1stfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 1stfval.d . . 3  |-  ( ph  ->  D  e.  Cat )
4 fvex 5742 . . . . . . 7  |-  ( Base `  c )  e.  _V
5 fvex 5742 . . . . . . 7  |-  ( Base `  d )  e.  _V
64, 5xpex 4990 . . . . . 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 5732 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  c
)  =  ( Base `  C ) )
10 simpr 448 . . . . . . . 8  |-  ( ( c  =  C  /\  d  =  D )  ->  d  =  D )
1110fveq2d 5732 . . . . . . 7  |-  ( ( c  =  C  /\  d  =  D )  ->  ( Base `  d
)  =  ( Base `  D ) )
129, 11xpeq12d 4903 . . . . . 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 2436 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
15 eqid 2436 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
1613, 14, 15xpcbas 14275 . . . . . . 7  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  (
Base `  T )
17 1stfval.b . . . . . . 7  |-  B  =  ( Base `  T
)
1816, 17eqtr4i 2459 . . . . . 6  |-  ( (
Base `  C )  X.  ( Base `  D
) )  =  B
1912, 18syl6eq 2484 . . . . 5  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ( Base `  c
)  X.  ( Base `  d ) )  =  B )
20 simpr 448 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2120reseq2d 5146 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 1st  |`  b )  =  ( 1st  |`  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 6099 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  ( C  X.c  D ) )
2524, 13syl6eqr 2486 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
c  X.c  d )  =  T )
2625fveq2d 5732 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  (  Hom  `  T )
)
27 1stfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  T
)
2826, 27syl6eqr 2486 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (  Hom  `  ( c  X.c  d ) )  =  H )
2928oveqd 6098 . . . . . . . 8  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x (  Hom  `  (
c  X.c  d ) ) y )  =  ( x H y ) )
3029reseq2d 5146 . . . . . . 7  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  ( 1st  |`  ( x (  Hom  `  ( c  X.c  d ) ) y ) )  =  ( 1st  |`  ( x H y ) ) )
3120, 20, 30mpt2eq123dv 6136 . . . . . 6  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  (
x  e.  b ,  y  e.  b  |->  ( 1st  |`  ( x
(  Hom  `  ( c  X.c  d ) ) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) )
3221, 31opeq12d 3992 . . . . 5  |-  ( ( ( c  =  C  /\  d  =  D )  /\  b  =  B )  ->  <. ( 1st  |`  b ) ,  ( x  e.  b ,  y  e.  b 
|->  ( 1st  |`  (
x (  Hom  `  (
c  X.c  d ) ) y ) ) ) >.  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
337, 19, 32csbied2 3294 . . . 4  |-  ( ( c  =  C  /\  d  =  D )  ->  [_ ( ( Base `  c )  X.  ( Base `  d ) )  /  b ]_ <. ( 1st  |`  b ) ,  ( x  e.  b ,  y  e.  b  |->  ( 1st  |`  (
x (  Hom  `  (
c  X.c  d ) ) y ) ) ) >.  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
34 df-1stf 14270 . . . 4  |-  1stF  =  (
c  e.  Cat , 
d  e.  Cat  |->  [_ ( ( Base `  c
)  X.  ( Base `  d ) )  / 
b ]_ <. ( 1st  |`  b
) ,  ( x  e.  b ,  y  e.  b  |->  ( 1st  |`  ( x (  Hom  `  ( c  X.c  d ) ) y ) ) ) >. )
35 opex 4427 . . . 4  |-  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >.  e.  _V
3633, 34, 35ovmpt2a 6204 . . 3  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C  1stF  D )  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
372, 3, 36syl2anc 643 . 2  |-  ( ph  ->  ( C  1stF  D )  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
381, 37syl5eq 2480 1  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251   <.cop 3817    X. cxp 4876    |` cres 4880   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   Basecbs 13469    Hom chom 13540   Catccat 13889    X.c cxpc 14265    1stF c1stf 14266
This theorem is referenced by:  1stf1  14289  1stf2  14290  1stfcl  14294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-hom 13553  df-cco 13554  df-xpc 14269  df-1stf 14270
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