MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stf1 Structured version   Unicode version

Theorem 1stf1 15090
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 15089 . . . 4  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
8 fo1st 6682 . . . . . . 7  |-  1st : _V -onto-> _V
9 fofun 5705 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
108, 9ax-mp 5 . . . . . 6  |-  Fun  1st
11 fvex 5785 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2532 . . . . . 6  |-  B  e. 
_V
13 resfunexg 6028 . . . . . 6  |-  ( ( Fun  1st  /\  B  e. 
_V )  ->  ( 1st  |`  B )  e. 
_V )
1410, 12, 13mp2an 672 . . . . 5  |-  ( 1st  |`  B )  e.  _V
1512, 12mpt2ex 6736 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6673 . . . 4  |-  ( P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >.  ->  ( 1st `  P )  =  ( 1st  |`  B ) )
177, 16syl 16 . . 3  |-  ( ph  ->  ( 1st `  P
)  =  ( 1st  |`  B ) )
1817fveq1d 5777 . 2  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( ( 1st  |`  B ) `  R
) )
19 1stf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5789 . . 3  |-  ( R  e.  B  ->  (
( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2119, 20syl 16 . 2  |-  ( ph  ->  ( ( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2218, 21eqtrd 2490 1  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( 1st `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   _Vcvv 3054   <.cop 3967    |` cres 4926   Fun wfun 5496   -onto->wfo 5500   ` cfv 5502  (class class class)co 6176    |-> cmpt2 6178   1stc1st 6661   Basecbs 14262   Hom chom 14337   Catccat 14690    X.c cxpc 15066    1stF c1stf 15067
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-uz 10949  df-fz 11525  df-struct 14264  df-ndx 14265  df-slot 14266  df-base 14267  df-hom 14350  df-cco 14351  df-xpc 15070  df-1stf 15071
This theorem is referenced by:  prf1st  15102  1st2ndprf  15104  uncf1  15134  uncf2  15135  diag11  15141  yonedalem21  15171  yonedalem22  15176
  Copyright terms: Public domain W3C validator