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Theorem caofinvl 6332
Description: Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofinv.3  |-  ( ph  ->  B  e.  W )
caofinv.4  |-  ( ph  ->  N : S --> S )
caofinv.5  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
caofinvl.6  |-  ( (
ph  /\  x  e.  S )  ->  (
( N `  x
) R x )  =  B )
Assertion
Ref Expression
caofinvl  |-  ( ph  ->  ( G  o F R F )  =  ( A  X.  { B } ) )
Distinct variable groups:    x, B    x, F    x, G    ph, x    x, R    x, S    v, A    v, F, x    x, N, v    v, S    ph, v
Allowed substitution hints:    A( x)    B( v)    R( v)    G( v)    V( x, v)    W( x, v)

Proof of Theorem caofinvl
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . . . 4  |-  ( ph  ->  A  e.  V )
2 caofinv.4 . . . . . . . . 9  |-  ( ph  ->  N : S --> S )
32adantr 453 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  N : S --> S )
4 caofref.2 . . . . . . . . 9  |-  ( ph  ->  F : A --> S )
54ffvelrnda 5871 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5872 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2437 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5894 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5581 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 225 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5871 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5871 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
14 fvex 5743 . . . . . . 7  |-  ( N `
 ( F `  v ) )  e. 
_V
1514, 7fnmpti 5574 . . . . . 6  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  Fn  A
169fneq1d 5537 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1715, 16mpbiri 226 . . . . 5  |-  ( ph  ->  G  Fn  A )
18 dffn5 5773 . . . . 5  |-  ( G  Fn  A  <->  G  =  ( w  e.  A  |->  ( G `  w
) ) )
1917, 18sylib 190 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
204feqmptd 5780 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
211, 12, 13, 19, 20offval2 6323 . . 3  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
229fveq1d 5731 . . . . . . 7  |-  ( ph  ->  ( G `  w
)  =  ( ( v  e.  A  |->  ( N `  ( F `
 v ) ) ) `  w ) )
23 fveq2 5729 . . . . . . . . 9  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2423fveq2d 5733 . . . . . . . 8  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
25 fvex 5743 . . . . . . . 8  |-  ( N `
 ( F `  w ) )  e. 
_V
2624, 7, 25fvmpt 5807 . . . . . . 7  |-  ( w  e.  A  ->  (
( v  e.  A  |->  ( N `  ( F `  v )
) ) `  w
)  =  ( N `
 ( F `  w ) ) )
2722, 26sylan9eq 2489 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
2827oveq1d 6097 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
29 caofinvl.6 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  (
( N `  x
) R x )  =  B )
3029ralrimiva 2790 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
3130adantr 453 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
32 fveq2 5729 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
33 id 21 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3432, 33oveq12d 6100 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
3534eqeq1d 2445 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( ( N `  x ) R x )  =  B  <->  ( ( N `  ( F `  w ) ) R ( F `  w
) )  =  B ) )
3635rspcva 3051 . . . . . 6  |-  ( ( ( F `  w
)  e.  S  /\  A. x  e.  S  ( ( N `  x
) R x )  =  B )  -> 
( ( N `  ( F `  w ) ) R ( F `
 w ) )  =  B )
3713, 31, 36syl2anc 644 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
3828, 37eqtrd 2469 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
3938mpteq2dva 4296 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4021, 39eqtrd 2469 . 2  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  B ) )
41 fconstmpt 4922 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4240, 41syl6eqr 2487 1  |-  ( ph  ->  ( G  o F R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   {csn 3815    e. cmpt 4267    X. cxp 4877    Fn wfn 5450   -->wf 5451   ` cfv 5455  (class class class)co 6082    o Fcof 6304
This theorem is referenced by:  grpvlinv  27428  lflnegl  29875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306
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