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Theorem 1stf1 13966
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 )
1stfval.p  |-  P  =  ( C  1stF  D )
1stf1.p  |-  ( ph  ->  R  e.  B )
Assertion
Ref Expression
1stf1  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( 1st `  R
) )

Proof of Theorem 1stf1
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 1stfval.p . . . . 5  |-  P  =  ( C  1stF  D )
71, 2, 3, 4, 5, 61stfval 13965 . . . 4  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
8 fo1st 6139 . . . . . . 7  |-  1st : _V -onto-> _V
9 fofun 5452 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
108, 9ax-mp 8 . . . . . 6  |-  Fun  1st
11 fvex 5539 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2353 . . . . . 6  |-  B  e. 
_V
13 resfunexg 5737 . . . . . 6  |-  ( ( Fun  1st  /\  B  e. 
_V )  ->  ( 1st  |`  B )  e. 
_V )
1410, 12, 13mp2an 653 . . . . 5  |-  ( 1st  |`  B )  e.  _V
1512, 12mpt2ex 6198 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6130 . . . 4  |-  ( P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >.  ->  ( 1st `  P )  =  ( 1st  |`  B ) )
177, 16syl 15 . . 3  |-  ( ph  ->  ( 1st `  P
)  =  ( 1st  |`  B ) )
1817fveq1d 5527 . 2  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( ( 1st  |`  B ) `  R
) )
19 1stf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5542 . . 3  |-  ( R  e.  B  ->  (
( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2119, 20syl 15 . 2  |-  ( ph  ->  ( ( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2218, 21eqtrd 2315 1  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( 1st `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    |` cres 4691   Fun wfun 5249   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   Basecbs 13148    Hom chom 13219   Catccat 13566    X.c cxpc 13942    1stF c1stf 13943
This theorem is referenced by:  prf1st  13978  1st2ndprf  13980  uncf1  14010  uncf2  14011  diag11  14017  yonedalem21  14047  yonedalem22  14052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-xpc 13946  df-1stf 13947
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