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Theorem 1stf1 13982
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 13981 . . . 4  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
8 fo1st 6155 . . . . . . 7  |-  1st : _V -onto-> _V
9 fofun 5468 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
108, 9ax-mp 8 . . . . . 6  |-  Fun  1st
11 fvex 5555 . . . . . . 7  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2366 . . . . . 6  |-  B  e. 
_V
13 resfunexg 5753 . . . . . 6  |-  ( ( Fun  1st  /\  B  e. 
_V )  ->  ( 1st  |`  B )  e. 
_V )
1410, 12, 13mp2an 653 . . . . 5  |-  ( 1st  |`  B )  e.  _V
1512, 12mpt2ex 6214 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) )  e. 
_V
1614, 15op1std 6146 . . . 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 5543 . 2  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( ( 1st  |`  B ) `  R
) )
19 1stf1.p . . 3  |-  ( ph  ->  R  e.  B )
20 fvres 5558 . . 3  |-  ( R  e.  B  ->  (
( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2119, 20syl 15 . 2  |-  ( ph  ->  ( ( 1st  |`  B ) `
 R )  =  ( 1st `  R
) )
2218, 21eqtrd 2328 1  |-  ( ph  ->  ( ( 1st `  P
) `  R )  =  ( 1st `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    |` cres 4707   Fun wfun 5265   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   Basecbs 13164    Hom chom 13235   Catccat 13582    X.c cxpc 13958    1stF c1stf 13959
This theorem is referenced by:  prf1st  13994  1st2ndprf  13996  uncf1  14026  uncf2  14027  diag11  14033  yonedalem21  14063  yonedalem22  14068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-xpc 13962  df-1stf 13963
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