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Theorem idaf 14105
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 4341 . . 3  |-  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.  e.  _V
21a1i 10 . 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 2366 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
73, 4, 5, 6idafval 14099 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  ( ( Id `  C ) `
 x ) >.
) )
8 idaf.a . . . 4  |-  A  =  (Nat `  C )
9 eqid 2366 . . . 4  |-  (Homa `  C
)  =  (Homa `  C
)
108, 9homarw 14088 . . 3  |-  ( x (Homa
`  C ) x )  C_  A
115adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  C  e.  Cat )
12 simpr 447 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  x  e.  B )
133, 4, 11, 12, 9idahom 14102 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  ( x (Homa `  C
) x ) )
1410, 13sseldi 3264 . 2  |-  ( (
ph  /\  x  e.  B )  ->  (
I `  x )  e.  A )
152, 7, 14fmpt2d 5799 1  |-  ( ph  ->  I : B --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   <.cotp 3733   -->wf 5354   ` cfv 5358  (class class class)co 5981   Basecbs 13356   Catccat 13776   Idccid 13777  Natcarw 14064  Homachoma 14065  Idacida 14095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-ot 3739  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-riota 6446  df-cat 13780  df-cid 13781  df-homa 14068  df-arw 14069  df-ida 14097
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