MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofid0r Structured version   Unicode version

Theorem caofid0r 6550
Description: Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofid0.3  |-  ( ph  ->  B  e.  W )
caofid0r.5  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  x )
Assertion
Ref Expression
caofid0r  |-  ( ph  ->  ( F  oF R ( A  X.  { B } ) )  =  F )
Distinct variable groups:    x, B    x, F    ph, x    x, R    x, S
Allowed substitution hints:    A( x)    V( x)    W( x)

Proof of Theorem caofid0r
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2  |-  ( ph  ->  A  e.  V )
2 caofref.2 . . 3  |-  ( ph  ->  F : A --> S )
3 ffn 5717 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ph  ->  F  Fn  A )
5 caofid0.3 . . 3  |-  ( ph  ->  B  e.  W )
6 fnconstg 5759 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
8 eqidd 2442 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
9 fvconst2g 6105 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
105, 9sylan 471 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
112ffvelrnda 6012 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
12 caofid0r.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  x )
1312ralrimiva 2855 . . . 4  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  x )
14 oveq1 6284 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
15 id 22 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1614, 15eqeq12d 2463 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  x  <->  ( ( F `  w ) R B )  =  ( F `  w ) ) )
1716rspccva 3193 . . . 4  |-  ( ( A. x  e.  S  ( x R B )  =  x  /\  ( F `  w )  e.  S )  -> 
( ( F `  w ) R B )  =  ( F `
 w ) )
1813, 17sylan 471 . . 3  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( ( F `  w ) R B )  =  ( F `  w ) )
1911, 18syldan 470 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  ( F `  w ) )
201, 4, 7, 4, 8, 10, 19offveq 6542 1  |-  ( ph  ->  ( F  oF R ( A  X.  { B } ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   {csn 4010    X. cxp 4983    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521
This theorem is referenced by:  psrlidm  17924  psrlidmOLD  17925  mndvrid  18763  lfl1sc  34511
  Copyright terms: Public domain W3C validator