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

Theorem idaf 15054
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 4668 . . 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 2454 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
73, 4, 5, 6idafval 15048 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.
) )
8 idaf.a . . . 4  |-  A  =  (Nat `  C )
9 eqid 2454 . . . 4  |-  (Homa `  C
)  =  (Homa `  C
)
108, 9homarw 15037 . . 3  |-  ( x (Homa
`  C ) x )  C_  A
115adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
12 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
133, 4, 11, 12, 9idahom 15051 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  ( x (Homa `  C
) x ) )
1410, 13sseldi 3465 . 2  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  A )
152, 7, 14fmpt2d 5985 1  |-  ( ph  ->  I : B --> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cotp 3996   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14296   Catccat 14725   Idccid 14726  Natcarw 15013  Homachoma 15014  Idacida 15044
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  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-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-cat 14729  df-cid 14730  df-homa 15017  df-arw 15018  df-ida 15046
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator