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Theorem caofid0l 6121
Description: Transfer a left 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 )
caofid0l.5  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  x )
Assertion
Ref Expression
caofid0l  |-  ( ph  ->  ( ( A  X.  { B } )  o F R F )  =  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 caofid0l
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2  |-  ( ph  ->  A  e.  V )
2 caofid0.3 . . 3  |-  ( ph  ->  B  e.  W )
3 fnconstg 5445 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
42, 3syl 15 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
5 caofref.2 . . 3  |-  ( ph  ->  F : A --> S )
6 ffn 5405 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
75, 6syl 15 . 2  |-  ( ph  ->  F  Fn  A )
8 fvconst2g 5743 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
92, 8sylan 457 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
10 eqidd 2297 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
11 ffvelrn 5679 . . . 4  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
125, 11sylan 457 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
13 caofid0l.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  x )
1413ralrimiva 2639 . . . 4  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  x )
15 oveq2 5882 . . . . . 6  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
16 id 19 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1715, 16eqeq12d 2310 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  x  <->  ( B R ( F `  w ) )  =  ( F `  w
) ) )
1817rspccva 2896 . . . 4  |-  ( ( A. x  e.  S  ( B R x )  =  x  /\  ( F `  w )  e.  S )  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
1914, 18sylan 457 . . 3  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
2012, 19syldan 456 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
211, 4, 7, 7, 9, 10, 20offveq 6114 1  |-  ( ph  ->  ( ( A  X.  { B } )  o F R F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  psr0lid  16156  psrlmod  16162  mndvlid  27551  mendlmod  27604  lfladd0l  29886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094
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