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Theorem caofid0r 6333
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  o F 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 5591 . . 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 5631 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
8 eqidd 2437 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
9 fvconst2g 5945 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
105, 9sylan 458 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
112ffvelrnda 5870 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
12 caofid0r.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  x )
1312ralrimiva 2789 . . . 4  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  x )
14 oveq1 6088 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
15 id 20 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1614, 15eqeq12d 2450 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  x  <->  ( ( F `  w ) R B )  =  ( F `  w ) ) )
1716rspccva 3051 . . . 4  |-  ( ( A. x  e.  S  ( x R B )  =  x  /\  ( F `  w )  e.  S )  -> 
( ( F `  w ) R B )  =  ( F `
 w ) )
1813, 17sylan 458 . . 3  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( ( F `  w ) R B )  =  ( F `  w ) )
1911, 18syldan 457 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  ( F `  w ) )
201, 4, 7, 4, 8, 10, 19offveq 6325 1  |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {csn 3814    X. cxp 4876    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303
This theorem is referenced by:  psrlidm  16467  mndvrid  27426  lfl1sc  29882
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
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