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Theorem 1stf2 15786
Description: Value of the first projection on a morphism. (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 )
1stf2.p  |-  ( ph  ->  S  e.  B )
Assertion
Ref Expression
1stf2  |-  ( ph  ->  ( R ( 2nd `  P ) S )  =  ( 1st  |`  ( R H S ) ) )

Proof of Theorem 1stf2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . 4  |-  T  =  ( C  X.c  D )
2 1stfval.b . . . 4  |-  B  =  ( Base `  T
)
3 1stfval.h . . . 4  |-  H  =  ( Hom  `  T
)
4 1stfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 1stfval.d . . . 4  |-  ( ph  ->  D  e.  Cat )
6 1stfval.p . . . 4  |-  P  =  ( C  1stF  D )
71, 2, 3, 4, 5, 61stfval 15784 . . 3  |-  ( ph  ->  P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >. )
8 fo1st 6804 . . . . . 6  |-  1st : _V -onto-> _V
9 fofun 5779 . . . . . 6  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
108, 9ax-mp 5 . . . . 5  |-  Fun  1st
11 fvex 5859 . . . . . 6  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2486 . . . . 5  |-  B  e. 
_V
13 resfunexg 6118 . . . . 5  |-  ( ( Fun  1st  /\  B  e. 
_V )  ->  ( 1st  |`  B )  e. 
_V )
1410, 12, 13mp2an 670 . . . 4  |-  ( 1st  |`  B )  e.  _V
1512, 12mpt2ex 6861 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) )  e. 
_V
1614, 15op2ndd 6795 . . 3  |-  ( P  =  <. ( 1st  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) >.  ->  ( 2nd `  P )  =  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  (
x H y ) ) ) )
177, 16syl 17 . 2  |-  ( ph  ->  ( 2nd `  P
)  =  ( x  e.  B ,  y  e.  B  |->  ( 1st  |`  ( x H y ) ) ) )
18 simprl 756 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  ->  x  =  R )
19 simprr 758 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
y  =  S )
2018, 19oveq12d 6296 . . 3  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
( x H y )  =  ( R H S ) )
2120reseq2d 5094 . 2  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
( 1st  |`  ( x H y ) )  =  ( 1st  |`  ( R H S ) ) )
22 1stf1.p . 2  |-  ( ph  ->  R  e.  B )
23 1stf2.p . 2  |-  ( ph  ->  S  e.  B )
24 ovex 6306 . . . 4  |-  ( R H S )  e. 
_V
25 resfunexg 6118 . . . 4  |-  ( ( Fun  1st  /\  ( R H S )  e. 
_V )  ->  ( 1st  |`  ( R H S ) )  e. 
_V )
2610, 24, 25mp2an 670 . . 3  |-  ( 1st  |`  ( R H S ) )  e.  _V
2726a1i 11 . 2  |-  ( ph  ->  ( 1st  |`  ( R H S ) )  e.  _V )
2817, 21, 22, 23, 27ovmpt2d 6411 1  |-  ( ph  ->  ( R ( 2nd `  P ) S )  =  ( 1st  |`  ( R H S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   <.cop 3978    |` cres 4825   Fun wfun 5563   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   Hom chom 14920   Catccat 15278    X.c cxpc 15761    1stF c1stf 15762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-hom 14933  df-cco 14934  df-xpc 15765  df-1stf 15766
This theorem is referenced by:  1stfcl  15790  prf1st  15797  1st2ndprf  15799  uncf2  15830  diag12  15837  diag2  15838  yonedalem22  15871
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