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Theorem idaf 14249
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 4457 . . 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 2442 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
73, 4, 5, 6idafval 14243 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.
) )
8 idaf.a . . . 4  |-  A  =  (Nat `  C )
9 eqid 2442 . . . 4  |-  (Homa `  C
)  =  (Homa `  C
)
108, 9homarw 14232 . . 3  |-  ( x (Homa
`  C ) x )  C_  A
115adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
12 simpr 449 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
133, 4, 11, 12, 9idahom 14246 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  ( x (Homa `  C
) x ) )
1410, 13sseldi 3332 . 2  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  A )
152, 7, 14fmpt2d 5927 1  |-  ( ph  ->  I : B --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   <.cotp 3842   -->wf 5479   ` cfv 5483  (class class class)co 6110   Basecbs 13500   Catccat 13920   Idccid 13921  Natcarw 14208  Homachoma 14209  Idacida 14239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-ot 3848  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-riota 6578  df-cat 13924  df-cid 13925  df-homa 14212  df-arw 14213  df-ida 14241
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