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Theorem caofinvl 5972
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 5523 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5524 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2117 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5542 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5229 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 166 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5523 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5523 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
146ralrimiva 2482 . . . . . . 7  |-  ( ph  ->  A. v  e.  A  ( N `  ( F `
 v ) )  e.  S )
157fnmpt 5219 . . . . . . 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 5183 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1816, 17mpbird 166 . . . . 5  |-  ( ph  ->  G  Fn  A )
19 dffn5im 5435 . . . . 5  |-  ( G  Fn  A  ->  G  =  ( w  e.  A  |->  ( G `  w ) ) )
2018, 19syl 14 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
214feqmptd 5442 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
221, 12, 13, 20, 21offval2 5965 . . 3  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
239fveq1d 5391 . . . . . . . 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 5524 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( N `  ( F `  w ) )  e.  S )
28 fveq2 5389 . . . . . . . . . 10  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2928fveq2d 5393 . . . . . . . . 9  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
3029, 7fvmptg 5465 . . . . . . . 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 2150 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3332oveq1d 5757 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
34 fveq2 5389 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
35 id 19 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3634, 35oveq12d 5760 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
3736eqeq1d 2126 . . . . . 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 2482 . . . . . . 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 2768 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4233, 41eqtrd 2150 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4342mpteq2dva 3988 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4422, 43eqtrd 2150 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  B ) )
45 fconstmpt 4556 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4644, 45syl6eqr 2168 1  |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393   {csn 3497    |-> cmpt 3959    X. cxp 4507    Fn wfn 5088   -->wf 5089   ` cfv 5093  (class class class)co 5742    oFcof 5948
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-of 5950
This theorem is referenced by: (None)
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