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Theorem caofinvl 5997
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  oF 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 274 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  N : S --> S )
4 caofref.2 . . . . . . . . 9  |-  ( ph  ->  F : A --> S )
54ffvelrnda 5548 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5549 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2137 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5567 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5254 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 166 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5548 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5548 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
146ralrimiva 2503 . . . . . . 7  |-  ( ph  ->  A. v  e.  A  ( N `  ( F `
 v ) )  e.  S )
157fnmpt 5244 . . . . . . 7  |-  ( A. v  e.  A  ( N `  ( F `  v ) )  e.  S  ->  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  Fn  A )
1614, 15syl 14 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) )  Fn  A
)
179fneq1d 5208 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1816, 17mpbird 166 . . . . 5  |-  ( ph  ->  G  Fn  A )
19 dffn5im 5460 . . . . 5  |-  ( G  Fn  A  ->  G  =  ( w  e.  A  |->  ( G `  w ) ) )
2018, 19syl 14 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
214feqmptd 5467 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
221, 12, 13, 20, 21offval2 5990 . . 3  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
239fveq1d 5416 . . . . . . . 8  |-  ( ph  ->  ( G `  w
)  =  ( ( v  e.  A  |->  ( N `  ( F `
 v ) ) ) `  w ) )
2423adantr 274 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) `
 w ) )
25 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
262adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  N : S --> S )
2726, 13ffvelrnd 5549 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( N `  ( F `  w ) )  e.  S )
28 fveq2 5414 . . . . . . . . . 10  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2928fveq2d 5418 . . . . . . . . 9  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
3029, 7fvmptg 5490 . . . . . . . 8  |-  ( ( w  e.  A  /\  ( N `  ( F `
 w ) )  e.  S )  -> 
( ( v  e.  A  |->  ( N `  ( F `  v ) ) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3125, 27, 30syl2anc 408 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  (
( v  e.  A  |->  ( N `  ( F `  v )
) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3224, 31eqtrd 2170 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3332oveq1d 5782 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
34 fveq2 5414 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
35 id 19 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3634, 35oveq12d 5785 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
3736eqeq1d 2146 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( ( N `  x ) R x )  =  B  <->  ( ( N `  ( F `  w ) ) R ( F `  w
) )  =  B ) )
38 caofinvl.6 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  (
( N `  x
) R x )  =  B )
3938ralrimiva 2503 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
4039adantr 274 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
4137, 40, 13rspcdva 2789 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4233, 41eqtrd 2170 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4342mpteq2dva 4013 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4422, 43eqtrd 2170 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  B ) )
45 fconstmpt 4581 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4644, 45syl6eqr 2188 1  |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2414   {csn 3522    |-> cmpt 3984    X. cxp 4532    Fn wfn 5113   -->wf 5114   ` cfv 5118  (class class class)co 5767    oFcof 5973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-of 5975
This theorem is referenced by: (None)
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