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Theorem caofid0l 6105
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 5429 . . 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 5389 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
75, 6syl 15 . 2  |-  ( ph  ->  F  Fn  A )
8 fvconst2g 5727 . . 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 2284 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
11 ffvelrn 5663 . . . 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 2626 . . . 4  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  x )
15 oveq2 5866 . . . . . 6  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
16 id 19 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1715, 16eqeq12d 2297 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  x  <->  ( B R ( F `  w ) )  =  ( F `  w
) ) )
1817rspccva 2883 . . . 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 6098 1  |-  ( ph  ->  ( ( A  X.  { B } )  o F R F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  psr0lid  16140  psrlmod  16146  mndvlid  27448  mendlmod  27501  lfladd0l  29264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078
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