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Theorem 2ndf2 15126
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 15124 . . 3  |-  ( ph  ->  Q  =  <. ( 2nd  |`  B ) ,  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  (
x H y ) ) ) >. )
8 fo2nd 6708 . . . . . 6  |-  2nd : _V -onto-> _V
9 fofun 5730 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
108, 9ax-mp 5 . . . . 5  |-  Fun  2nd
11 fvex 5810 . . . . . 6  |-  ( Base `  T )  e.  _V
122, 11eqeltri 2538 . . . . 5  |-  B  e. 
_V
13 resfunexg 6053 . . . . 5  |-  ( ( Fun  2nd  /\  B  e. 
_V )  ->  ( 2nd  |`  B )  e. 
_V )
1410, 12, 13mp2an 672 . . . 4  |-  ( 2nd  |`  B )  e.  _V
1512, 12mpt2ex 6761 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( 2nd  |`  ( x H y ) ) )  e. 
_V
1614, 15op2ndd 6699 . . 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 755 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  ->  x  =  R )
19 simprr 756 . . . 4  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
y  =  S )
2018, 19oveq12d 6219 . . 3  |-  ( (
ph  /\  ( x  =  R  /\  y  =  S ) )  -> 
( x H y )  =  ( R H S ) )
2120reseq2d 5219 . 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 6226 . . . 4  |-  ( R H S )  e. 
_V
25 resfunexg 6053 . . . 4  |-  ( ( Fun  2nd  /\  ( R H S )  e. 
_V )  ->  ( 2nd  |`  ( R H S ) )  e. 
_V )
2610, 24, 25mp2an 672 . . 3  |-  ( 2nd  |`  ( R H S ) )  e.  _V
2726a1i 11 . 2  |-  ( ph  ->  ( 2nd  |`  ( R H S ) )  e.  _V )
2817, 21, 22, 23, 27ovmpt2d 6329 1  |-  ( ph  ->  ( R ( 2nd `  Q ) S )  =  ( 2nd  |`  ( R H S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3992    |` cres 4951   Fun wfun 5521   -onto->wfo 5525   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   2ndc2nd 6687   Basecbs 14293   Hom chom 14369   Catccat 14722    X.c cxpc 15098    2ndF c2ndf 15100
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-hom 14382  df-cco 14383  df-xpc 15102  df-2ndf 15104
This theorem is referenced by:  2ndfcl  15128  prf2nd  15135  1st2ndprf  15136  uncf2  15167  curf2ndf  15177  yonedalem22  15208
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