MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idaf Unicode version

Theorem idaf 14173
Description: The identity arrow function is a function from objects to arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i  |-  I  =  (Ida
`  C )
idafval.b  |-  B  =  ( Base `  C
)
idafval.c  |-  ( ph  ->  C  e.  Cat )
idaf.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
idaf  |-  ( ph  ->  I : B --> A )

Proof of Theorem idaf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 otex 4388 . . 3  |-  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.  e.  _V
21a1i 11 . 2  |-  ( (
ph  /\  x  e.  B )  ->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.  e.  _V )
3 idafval.i . . 3  |-  I  =  (Ida
`  C )
4 idafval.b . . 3  |-  B  =  ( Base `  C
)
5 idafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
6 eqid 2404 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
73, 4, 5, 6idafval 14167 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.
) )
8 idaf.a . . . 4  |-  A  =  (Nat `  C )
9 eqid 2404 . . . 4  |-  (Homa `  C
)  =  (Homa `  C
)
108, 9homarw 14156 . . 3  |-  ( x (Homa
`  C ) x )  C_  A
115adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
12 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
133, 4, 11, 12, 9idahom 14170 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  ( x (Homa `  C
) x ) )
1410, 13sseldi 3306 . 2  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  A )
152, 7, 14fmpt2d 5857 1  |-  ( ph  ->  I : B --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cotp 3778   -->wf 5409   ` cfv 5413  (class class class)co 6040   Basecbs 13424   Catccat 13844   Idccid 13845  Natcarw 14132  Homachoma 14133  Idacida 14163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-ot 3784  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-riota 6508  df-cat 13848  df-cid 13849  df-homa 14136  df-arw 14137  df-ida 14165
  Copyright terms: Public domain W3C validator