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Theorem caofinvl 5758
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 5328 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5329 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2082 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5348 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5059 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 165 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5328 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5328 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
146ralrimiva 2435 . . . . . . 7  |-  ( ph  ->  A. v  e.  A  ( N `  ( F `
 v ) )  e.  S )
157fnmpt 5050 . . . . . . 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 5014 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1816, 17mpbird 165 . . . . 5  |-  ( ph  ->  G  Fn  A )
19 dffn5im 5245 . . . . 5  |-  ( G  Fn  A  ->  G  =  ( w  e.  A  |->  ( G `  w ) ) )
2018, 19syl 14 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
214feqmptd 5252 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
221, 12, 13, 20, 21offval2 5751 . . 3  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
239fveq1d 5205 . . . . . . . 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 5329 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( N `  ( F `  w ) )  e.  S )
28 fveq2 5203 . . . . . . . . . 10  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2928fveq2d 5207 . . . . . . . . 9  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
3029, 7fvmptg 5274 . . . . . . . 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 2114 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3332oveq1d 5552 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
34 caofinvl.6 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  (
( N `  x
) R x )  =  B )
3534ralrimiva 2435 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
3635adantr 270 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
37 fveq2 5203 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
38 id 19 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3937, 38oveq12d 5555 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
4039eqeq1d 2090 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( ( N `  x ) R x )  =  B  <->  ( ( N `  ( F `  w ) ) R ( F `  w
) )  =  B ) )
4140rspcva 2700 . . . . . 6  |-  ( ( ( F `  w
)  e.  S  /\  A. x  e.  S  ( ( N `  x
) R x )  =  B )  -> 
( ( N `  ( F `  w ) ) R ( F `
 w ) )  =  B )
4213, 36, 41syl2anc 403 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4333, 42eqtrd 2114 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4443mpteq2dva 3870 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4522, 44eqtrd 2114 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  B ) )
46 fconstmpt 4407 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4745, 46syl6eqr 2132 1  |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   {csn 3400    |-> cmpt 3841    X. cxp 4363    Fn wfn 4921   -->wf 4922   ` cfv 4926  (class class class)co 5537    oFcof 5735
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-setind 4282
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-of 5737
This theorem is referenced by: (None)
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