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Theorem caofinvl 6104
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 451 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  N : S --> S )
4 caofref.2 . . . . . . . . 9  |-  ( ph  ->  F : A --> S )
5 ffvelrn 5663 . . . . . . . . 9  |-  ( ( F : A --> S  /\  v  e.  A )  ->  ( F `  v
)  e.  S )
64, 5sylan 457 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
7 ffvelrn 5663 . . . . . . . 8  |-  ( ( N : S --> S  /\  ( F `  v )  e.  S )  -> 
( N `  ( F `  v )
)  e.  S )
83, 6, 7syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
9 eqid 2283 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
108, 9fmptd 5684 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
11 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
1211feq1d 5379 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
1310, 12mpbird 223 . . . . 5  |-  ( ph  ->  G : A --> S )
14 ffvelrn 5663 . . . . 5  |-  ( ( G : A --> S  /\  w  e.  A )  ->  ( G `  w
)  e.  S )
1513, 14sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
16 ffvelrn 5663 . . . . 5  |-  ( ( F : A --> S  /\  w  e.  A )  ->  ( F `  w
)  e.  S )
174, 16sylan 457 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
18 fvex 5539 . . . . . . 7  |-  ( N `
 ( F `  v ) )  e. 
_V
1918, 9fnmpti 5372 . . . . . 6  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  Fn  A
2011fneq1d 5335 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
2119, 20mpbiri 224 . . . . 5  |-  ( ph  ->  G  Fn  A )
22 dffn5 5568 . . . . 5  |-  ( G  Fn  A  <->  G  =  ( w  e.  A  |->  ( G `  w
) ) )
2321, 22sylib 188 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
244feqmptd 5575 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
251, 15, 17, 23, 24offval2 6095 . . 3  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
2611fveq1d 5527 . . . . . . 7  |-  ( ph  ->  ( G `  w
)  =  ( ( v  e.  A  |->  ( N `  ( F `
 v ) ) ) `  w ) )
27 fveq2 5525 . . . . . . . . 9  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2827fveq2d 5529 . . . . . . . 8  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
29 fvex 5539 . . . . . . . 8  |-  ( N `
 ( F `  w ) )  e. 
_V
3028, 9, 29fvmpt 5602 . . . . . . 7  |-  ( w  e.  A  ->  (
( v  e.  A  |->  ( N `  ( F `  v )
) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3126, 30sylan9eq 2335 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3231oveq1d 5873 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
33 caofinvl.6 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  (
( N `  x
) R x )  =  B )
3433ralrimiva 2626 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
3534adantr 451 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
36 fveq2 5525 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
37 id 19 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3836, 37oveq12d 5876 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
3938eqeq1d 2291 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( ( N `  x ) R x )  =  B  <->  ( ( N `  ( F `  w ) ) R ( F `  w
) )  =  B ) )
4039rspcva 2882 . . . . . 6  |-  ( ( ( F `  w
)  e.  S  /\  A. x  e.  S  ( ( N `  x
) R x )  =  B )  -> 
( ( N `  ( F `  w ) ) R ( F `
 w ) )  =  B )
4117, 35, 40syl2anc 642 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4232, 41eqtrd 2315 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4342mpteq2dva 4106 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4425, 43eqtrd 2315 . 2  |-  ( ph  ->  ( G  o F R F )  =  ( w  e.  A  |->  B ) )
45 fconstmpt 4732 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4644, 45syl6eqr 2333 1  |-  ( ph  ->  ( G  o F R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {csn 3640    e. cmpt 4077    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  grpvlinv  27450  lflnegl  29266
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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