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Theorem caofid0r 6458
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 5666 . . 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 5705 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
8 eqidd 2455 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
9 fvconst2g 6039 . . 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 5951 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
12 caofid0r.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  x )
1312ralrimiva 2829 . . . 4  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  x )
14 oveq1 6206 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
15 id 22 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1614, 15eqeq12d 2476 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  x  <->  ( ( F `  w ) R B )  =  ( F `  w ) ) )
1716rspccva 3176 . . . 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 6450 1  |-  ( ph  ->  ( F  oF R ( A  X.  { B } ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2798   {csn 3984    X. cxp 4945    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6199    oFcof 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429
This theorem is referenced by:  psrlidm  17596  psrlidmOLD  17597  mndvrid  18418  lfl1sc  33052
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