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

Theorem idafval 14204
Description: Value of the identity arrow function. (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 )
idafval.1  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
idafval  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Distinct variable groups:    x,  .1.    x, B    x, C    x, I    ph, x

Proof of Theorem idafval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 idafval.i . 2  |-  I  =  (Ida
`  C )
2 idafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5720 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 idafval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2485 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5720 . . . . . . . 8  |-  ( c  =  C  ->  ( Id `  c )  =  ( Id `  C
) )
7 idafval.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
86, 7syl6eqr 2485 . . . . . . 7  |-  ( c  =  C  ->  ( Id `  c )  =  .1.  )
98fveq1d 5722 . . . . . 6  |-  ( c  =  C  ->  (
( Id `  c
) `  x )  =  (  .1.  `  x
) )
109oteq3d 3990 . . . . 5  |-  ( c  =  C  ->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.  =  <. x ,  x ,  (  .1.  `  x
) >. )
115, 10mpteq12dv 4279 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c )  |->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
12 df-ida 14202 . . . 4  |- Ida  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c
)  |->  <. x ,  x ,  ( ( Id
`  c ) `  x ) >. )
)
13 fvex 5734 . . . . . 6  |-  ( Base `  C )  e.  _V
144, 13eqeltri 2505 . . . . 5  |-  B  e. 
_V
1514mptex 5958 . . . 4  |-  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )  e.  _V
1611, 12, 15fvmpt 5798 . . 3  |-  ( C  e.  Cat  ->  (Ida `  C
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
172, 16syl 16 . 2  |-  ( ph  ->  (Ida
`  C )  =  ( x  e.  B  |-> 
<. x ,  x ,  (  .1.  `  x
) >. ) )
181, 17syl5eq 2479 1  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cotp 3810    e. cmpt 4258   ` cfv 5446   Basecbs 13461   Catccat 13881   Idccid 13882  Idacida 14200
This theorem is referenced by:  idaval  14205  idaf  14210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ida 14202
  Copyright terms: Public domain W3C validator