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Theorem coapm 14937
Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coapm.o  |-  .x.  =  (compa `  C )
coapm.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
coapm  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )

Proof of Theorem coapm
Dummy variables  f 
g  h  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coapm.o . . . . . 6  |-  .x.  =  (compa `  C )
2 coapm.a . . . . . 6  |-  A  =  (Nat `  C )
3 eqid 2441 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 14930 . . . . 5  |-  .x.  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  C ) (coda `  g
) ) ( 2nd `  f ) ) >.
)
54mpt2fun 6190 . . . 4  |-  Fun  .x.
6 funfn 5445 . . . 4  |-  ( Fun 
.x. 
<-> 
.x.  Fn  dom  .x.  )
75, 6mpbi 208 . . 3  |-  .x.  Fn  dom  .x.
81, 2dmcoass 14932 . . . . . . . . 9  |-  dom  .x.  C_  ( A  X.  A
)
98sseli 3350 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  z  e.  ( A  X.  A ) )
10 1st2nd2 6611 . . . . . . . 8  |-  ( z  e.  ( A  X.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
119, 10syl 16 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
1211fveq2d 5693 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  (  .x.  `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. ) )
13 df-ov 6092 . . . . . 6  |-  ( ( 1st `  z ) 
.x.  ( 2nd `  z
) )  =  ( 
.x.  `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )
1412, 13syl6eqr 2491 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  =  ( ( 1st `  z )  .x.  ( 2nd `  z ) ) )
15 eqid 2441 . . . . . . 7  |-  (Homa `  C
)  =  (Homa `  C
)
162, 15homarw 14912 . . . . . 6  |-  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 1st `  z ) ) )  C_  A
17 id 22 . . . . . . . . . . . . 13  |-  ( z  e.  dom  .x.  ->  z  e.  dom  .x.  )
1811, 17eqeltrrd 2516 . . . . . . . . . . . 12  |-  ( z  e.  dom  .x.  ->  <.
( 1st `  z
) ,  ( 2nd `  z ) >.  e.  dom  .x.  )
19 df-br 4291 . . . . . . . . . . . 12  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  dom  .x.  )
2018, 19sylibr 212 . . . . . . . . . . 11  |-  ( z  e.  dom  .x.  ->  ( 1st `  z ) dom  .x.  ( 2nd `  z ) )
211, 2eldmcoa 14931 . . . . . . . . . . 11  |-  ( ( 1st `  z ) dom  .x.  ( 2nd `  z )  <->  ( ( 2nd `  z )  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda
`  ( 2nd `  z
) )  =  (domA `  ( 1st `  z ) ) ) )
2220, 21sylib 196 . . . . . . . . . 10  |-  ( z  e.  dom  .x.  ->  ( ( 2nd `  z
)  e.  A  /\  ( 1st `  z )  e.  A  /\  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z
) ) ) )
2322simp1d 1000 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  A )
242, 15arwhoma 14911 . . . . . . . . 9  |-  ( ( 2nd `  z )  e.  A  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 2nd `  z
) ) ) )
2523, 24syl 16 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (coda `  ( 2nd `  z ) ) ) )
2622simp3d 1002 . . . . . . . . 9  |-  ( z  e.  dom  .x.  ->  (coda `  ( 2nd `  z ) )  =  (domA `  ( 1st `  z ) ) )
2726oveq2d 6105 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (coda `  ( 2nd `  z ) ) )  =  ( (domA `  ( 2nd `  z ) ) (Homa
`  C ) (domA `  ( 1st `  z ) ) ) )
2825, 27eleqtrd 2517 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 2nd `  z )  e.  ( (domA `  ( 2nd `  z ) ) (Homa `  C ) (domA `  ( 1st `  z ) ) ) )
2922simp2d 1001 . . . . . . . 8  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  A )
302, 15arwhoma 14911 . . . . . . . 8  |-  ( ( 1st `  z )  e.  A  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3129, 30syl 16 . . . . . . 7  |-  ( z  e.  dom  .x.  ->  ( 1st `  z )  e.  ( (domA `  ( 1st `  z ) ) (Homa `  C ) (coda `  ( 1st `  z ) ) ) )
321, 15, 28, 31coahom 14936 . . . . . 6  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  ( (domA `  ( 2nd `  z
) ) (Homa `  C
) (coda
`  ( 1st `  z
) ) ) )
3316, 32sseldi 3352 . . . . 5  |-  ( z  e.  dom  .x.  ->  ( ( 1st `  z
)  .x.  ( 2nd `  z ) )  e.  A )
3414, 33eqeltrd 2515 . . . 4  |-  ( z  e.  dom  .x.  ->  ( 
.x.  `  z )  e.  A )
3534rgen 2779 . . 3  |-  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
36 ffnfv 5867 . . 3  |-  (  .x.  : dom  .x.  --> A  <->  (  .x.  Fn  dom  .x.  /\  A. z  e.  dom  .x.  (  .x.  `  z )  e.  A
) )
377, 35, 36mpbir2an 911 . 2  |-  .x.  : dom  .x.  --> A
38 fvex 5699 . . . 4  |-  (Nat `  C )  e.  _V
392, 38eqeltri 2511 . . 3  |-  A  e. 
_V
4039, 39xpex 6506 . . 3  |-  ( A  X.  A )  e. 
_V
4139, 40elpm2 7242 . 2  |-  (  .x.  e.  ( A  ^pm  ( A  X.  A ) )  <-> 
(  .x.  : dom  .x.  --> A  /\  dom  .x.  C_  ( A  X.  A ) ) )
4237, 8, 41mpbir2an 911 1  |-  .x.  e.  ( A  ^pm  ( A  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970    C_ wss 3326   <.cop 3881   <.cotp 3883   class class class wbr 4290    X. cxp 4836   dom cdm 4838   Fun wfun 5410    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   1stc1st 6573   2ndc2nd 6574    ^pm cpm 7213  compcco 14248  domAcdoma 14886  codaccoda 14887  Natcarw 14888  Homachoma 14889  compaccoa 14920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-ot 3884  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-pm 7215  df-cat 14604  df-doma 14890  df-coda 14891  df-homa 14892  df-arw 14893  df-coa 14922
This theorem is referenced by: (None)
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