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Theorem caofinvl 5869
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 270 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  N : S --> S )
4 caofref.2 . . . . . . . . 9  |-  ( ph  ->  F : A --> S )
54ffvelrnda 5428 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5429 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2088 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5446 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5143 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 165 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5428 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5428 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
146ralrimiva 2446 . . . . . . 7  |-  ( ph  ->  A. v  e.  A  ( N `  ( F `
 v ) )  e.  S )
157fnmpt 5134 . . . . . . 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 5098 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1816, 17mpbird 165 . . . . 5  |-  ( ph  ->  G  Fn  A )
19 dffn5im 5344 . . . . 5  |-  ( G  Fn  A  ->  G  =  ( w  e.  A  |->  ( G `  w ) ) )
2018, 19syl 14 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
214feqmptd 5351 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
221, 12, 13, 20, 21offval2 5862 . . 3  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
239fveq1d 5301 . . . . . . . 8  |-  ( ph  ->  ( G `  w
)  =  ( ( v  e.  A  |->  ( N `  ( F `
 v ) ) ) `  w ) )
2423adantr 270 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) `
 w ) )
25 simpr 108 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
262adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  N : S --> S )
2726, 13ffvelrnd 5429 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( N `  ( F `  w ) )  e.  S )
28 fveq2 5299 . . . . . . . . . 10  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2928fveq2d 5303 . . . . . . . . 9  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
3029, 7fvmptg 5374 . . . . . . . 8  |-  ( ( w  e.  A  /\  ( N `  ( F `
 w ) )  e.  S )  -> 
( ( v  e.  A  |->  ( N `  ( F `  v ) ) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3125, 27, 30syl2anc 403 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  (
( v  e.  A  |->  ( N `  ( F `  v )
) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3224, 31eqtrd 2120 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3332oveq1d 5659 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
34 fveq2 5299 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
35 id 19 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3634, 35oveq12d 5662 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
3736eqeq1d 2096 . . . . . 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 2446 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
4039adantr 270 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
4137, 40, 13rspcdva 2727 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4233, 41eqtrd 2120 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4342mpteq2dva 3926 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4422, 43eqtrd 2120 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  B ) )
45 fconstmpt 4481 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4644, 45syl6eqr 2138 1  |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   {csn 3444    |-> cmpt 3897    X. cxp 4434    Fn wfn 5005   -->wf 5006   ` cfv 5010  (class class class)co 5644    oFcof 5846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-setind 4351
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-of 5848
This theorem is referenced by: (None)
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