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Theorem 2ndf1 15339
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 15338 . . . 4  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
8 fo2nd 6816 . . . . . . 7  |-  2nd : _V -onto-> _V
9 fofun 5802 . . . . . . 7  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
108, 9ax-mp 5 . . . . . 6  |-  Fun  2nd
11 fvex 5882 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2551 . . . . . 6  |-  B  e. 
_V
13 resfunexg 6137 . . . . . 6  |-  ( ( Fun  2nd  /\  B  e. 
_V )  ->  ( 2nd  |`  B )  e. 
_V )
1410, 12, 13mp2an 672 . . . . 5  |-  ( 2nd  |`  B )  e.  _V
1512, 12mpt2ex 6872 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6805 . . . 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 5874 . 2  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( ( 2nd  |`  B ) `  R
) )
19 2ndf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5886 . . 3  |-  ( R  e.  B  ->  (
( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2119, 20syl 16 . 2  |-  ( ph  ->  ( ( 2nd  |`  B ) `
 R )  =  ( 2nd `  R
) )
2218, 21eqtrd 2508 1  |-  ( ph  ->  ( ( 1st `  Q
) `  R )  =  ( 2nd `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039    |` cres 5007   Fun wfun 5588   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   Basecbs 14507   Hom chom 14583   Catccat 14936    X.c cxpc 15312    2ndF c2ndf 15314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-xpc 15316  df-2ndf 15318
This theorem is referenced by:  prf2nd  15349  1st2ndprf  15350  uncf1  15380  uncf2  15381  curf2ndf  15391  yonedalem21  15417  yonedalem22  15422
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