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Theorem 2ndf1 15104
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 15103 . . . 4  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
8 fo2nd 6694 . . . . . . 7  |-  2nd : _V -onto-> _V
9 fofun 5716 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
108, 9ax-mp 5 . . . . . 6  |-  Fun  2nd
11 fvex 5796 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2533 . . . . . 6  |-  B  e. 
_V
13 resfunexg 6039 . . . . . 6  |-  ( ( Fun  2nd  /\  B  e. 
_V )  ->  ( 2nd  |`  B )  e. 
_V )
1410, 12, 13mp2an 672 . . . . 5  |-  ( 2nd  |`  B )  e.  _V
1512, 12mpt2ex 6747 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6684 . . . 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 5788 . 2  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( ( 2nd  |`  B ) `  R
) )
19 2ndf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5800 . . 3  |-  ( R  e.  B  ->  (
( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2119, 20syl 16 . 2  |-  ( ph  ->  ( ( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2218, 21eqtrd 2491 1  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( 2nd `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3065   <.cop 3978    |` cres 4937   Fun wfun 5507   -onto->wfo 5511   ` cfv 5513  (class class class)co 6187    |-> cmpt2 6189   1stc1st 6672   2ndc2nd 6673   Basecbs 14273   Hom chom 14348   Catccat 14701    X.c cxpc 15077    2ndF c2ndf 15079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-hom 14361  df-cco 14362  df-xpc 15081  df-2ndf 15083
This theorem is referenced by:  prf2nd  15114  1st2ndprf  15115  uncf1  15145  uncf2  15146  curf2ndf  15156  yonedalem21  15182  yonedalem22  15187
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