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Theorem caofid0l 6489
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 } )  oF 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 5698 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
42, 3syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
5 caofref.2 . . 3  |-  ( ph  ->  F : A --> S )
6 ffn 5656 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  F  Fn  A )
8 fvconst2g 6045 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
92, 8sylan 469 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
10 eqidd 2397 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
115ffvelrnda 5950 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
12 caofid0l.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  x )
1312ralrimiva 2810 . . . 4  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  x )
14 oveq2 6226 . . . . . 6  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
15 id 22 . . . . . 6  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
1614, 15eqeq12d 2418 . . . . 5  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  x  <->  ( B R ( F `  w ) )  =  ( F `  w
) ) )
1716rspccva 3151 . . . 4  |-  ( ( A. x  e.  S  ( B R x )  =  x  /\  ( F `  w )  e.  S )  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
1813, 17sylan 469 . . 3  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
1911, 18syldan 468 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( F `  w
) )
201, 4, 7, 7, 9, 10, 19offveq 6482 1  |-  ( ph  ->  ( ( A  X.  { B } )  oF R F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   {csn 3961    X. cxp 4928    Fn wfn 5508   -->wf 5509   ` cfv 5513  (class class class)co 6218    oFcof 6459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461
This theorem is referenced by:  psr0lid  18184  psrlmod  18190  mndvlid  19003  mendlmod  31350  lfladd0l  35251
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