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Theorem idaf 15666
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 4656 . . 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 2402 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
73, 4, 5, 6idafval 15660 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.
) )
8 idaf.a . . . 4  |-  A  =  (Nat `  C )
9 eqid 2402 . . . 4  |-  (Homa `  C
)  =  (Homa `  C
)
108, 9homarw 15649 . . 3  |-  ( x (Homa
`  C ) x )  C_  A
115adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
12 simpr 459 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
133, 4, 11, 12, 9idahom 15663 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  ( x (Homa `  C
) x ) )
1410, 13sseldi 3440 . 2  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  A )
152, 7, 14fmpt2d 6040 1  |-  ( ph  ->  I : B --> A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   <.cotp 3980   -->wf 5565   ` cfv 5569  (class class class)co 6278   Basecbs 14841   Catccat 15278   Idccid 15279  Natcarw 15625  Homachoma 15626  Idacida 15656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-ot 3981  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-cat 15282  df-cid 15283  df-homa 15629  df-arw 15630  df-ida 15658
This theorem is referenced by: (None)
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