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Theorem 2ndf2 14295
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 )
2ndfval.p  |-  Q  =  ( C  2ndF  D )
2ndf1.p  |-  ( ph  ->  R  e.  B )
2ndf2.p  |-  ( ph  ->  S  e.  B )
Assertion
Ref Expression
2ndf2  |-  ( ph  ->  ( R ( 2nd `  Q ) S )  =  ( 2nd  |`  ( R H S ) ) )

Proof of Theorem 2ndf2
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 2ndfval.p . . . 4  |-  Q  =  ( C  2ndF  D )
71, 2, 3, 4, 5, 62ndfval 14293 . . 3  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
8 fo2nd 6369 . . . . . 6  |-  2nd : _V -onto-> _V
9 fofun 5656 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
108, 9ax-mp 8 . . . . 5  |-  Fun  2nd
11 fvex 5744 . . . . . 6  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2508 . . . . 5  |-  B  e. 
_V
13 resfunexg 5959 . . . . 5  |-  ( ( Fun  2nd  /\  B  e. 
_V )  ->  ( 2nd  |`  B )  e. 
_V )
1410, 12, 13mp2an 655 . . . 4  |-  ( 2nd  |`  B )  e.  _V
1512, 12mpt2ex 6427 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )  e. 
_V
1614, 15op2ndd 6360 . . 3  |-  ( Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >.  ->  ( 2nd `  Q )  =  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) )
177, 16syl 16 . 2  |-  ( ph  ->  ( 2nd `  Q
)  =  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) ) )
18 simprl 734 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  ->  x  =  R )
19 simprr 735 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
y  =  S )
2018, 19oveq12d 6101 . . 3  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
( x H y )  =  ( R H S ) )
2120reseq2d 5148 . 2  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
( 2nd  |`  ( x H y ) )  =  ( 2nd  |`  ( R H S ) ) )
22 2ndf1.p . 2  |-  ( ph  ->  R  e.  B )
23 2ndf2.p . 2  |-  ( ph  ->  S  e.  B )
24 ovex 6108 . . . 4  |-  ( R H S )  e. 
_V
25 resfunexg 5959 . . . 4  |-  ( ( Fun  2nd  /\  ( R H S )  e. 
_V )  ->  ( 2nd  |`  ( R H S ) )  e. 
_V )
2610, 24, 25mp2an 655 . . 3  |-  ( 2nd  |`  ( R H S ) )  e.  _V
2726a1i 11 . 2  |-  ( ph  ->  ( 2nd  |`  ( R H S ) )  e.  _V )
2817, 21, 22, 23, 27ovmpt2d 6203 1  |-  ( ph  ->  ( R ( 2nd `  Q ) S )  =  ( 2nd  |`  ( R H S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    |` cres 4882   Fun wfun 5450   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   2ndc2nd 6350   Basecbs 13471    Hom chom 13542   Catccat 13891    X.c cxpc 14267    2ndF c2ndf 14269
This theorem is referenced by:  2ndfcl  14297  prf2nd  14304  1st2ndprf  14305  uncf2  14336  curf2ndf  14346  yonedalem22  14377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-xpc 14271  df-2ndf 14273
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