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Theorem 2ndf1 14280
Description: Value of the first projection on an object. (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 )
2ndf1.p  |-  ( ph  ->  R  e.  B )
Assertion
Ref Expression
2ndf1  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( 2nd `  R
) )

Proof of Theorem 2ndf1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . . 5  |-  T  =  ( C  X.c  D )
2 1stfval.b . . . . 5  |-  B  =  ( Base `  T
)
3 1stfval.h . . . . 5  |-  H  =  (  Hom  `  T
)
4 1stfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
5 1stfval.d . . . . 5  |-  ( ph  ->  D  e.  Cat )
6 2ndfval.p . . . . 5  |-  Q  =  ( C  2ndF  D )
71, 2, 3, 4, 5, 62ndfval 14279 . . . 4  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
8 fo2nd 6358 . . . . . . 7  |-  2nd : _V -onto-> _V
9 fofun 5645 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
108, 9ax-mp 8 . . . . . 6  |-  Fun  2nd
11 fvex 5733 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2505 . . . . . 6  |-  B  e. 
_V
13 resfunexg 5948 . . . . . 6  |-  ( ( Fun  2nd  /\  B  e. 
_V )  ->  ( 2nd  |`  B )  e. 
_V )
1410, 12, 13mp2an 654 . . . . 5  |-  ( 2nd  |`  B )  e.  _V
1512, 12mpt2ex 6416 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6348 . . . 4  |-  ( Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  ->  ( 1st `  Q )  =  ( 2nd  |`  B ) )
177, 16syl 16 . . 3  |-  ( ph  ->  ( 1st `  Q
)  =  ( 2nd  |`  B ) )
1817fveq1d 5721 . 2  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( ( 2nd  |`  B ) `  R
) )
19 2ndf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5736 . . 3  |-  ( R  e.  B  ->  (
( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2119, 20syl 16 . 2  |-  ( ph  ->  ( ( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2218, 21eqtrd 2467 1  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( 2nd `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    |` cres 4871   Fun wfun 5439   -onto->wfo 5443   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   1stc1st 6338   2ndc2nd 6339   Basecbs 13457    Hom chom 13528   Catccat 13877    X.c cxpc 14253    2ndF c2ndf 14255
This theorem is referenced by:  prf2nd  14290  1st2ndprf  14291  uncf1  14321  uncf2  14322  curf2ndf  14332  yonedalem21  14358  yonedalem22  14363
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-hom 13541  df-cco 13542  df-xpc 14257  df-2ndf 14259
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