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Theorem caofinvl 5733
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 261 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  N : S --> S )
4 caofref.2 . . . . . . . . 9  |-  ( ph  ->  F : A --> S )
54ffvelrnda 5302 . . . . . . . 8  |-  ( (
ph  /\  v  e.  A )  ->  ( F `  v )  e.  S )
63, 5ffvelrnd 5303 . . . . . . 7  |-  ( (
ph  /\  v  e.  A )  ->  ( N `  ( F `  v ) )  e.  S )
7 eqid 2040 . . . . . . 7  |-  ( v  e.  A  |->  ( N `
 ( F `  v ) ) )  =  ( v  e.  A  |->  ( N `  ( F `  v ) ) )
86, 7fmptd 5322 . . . . . 6  |-  ( ph  ->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S )
9 caofinv.5 . . . . . . 7  |-  ( ph  ->  G  =  ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) )
109feq1d 5034 . . . . . 6  |-  ( ph  ->  ( G : A --> S 
<->  ( v  e.  A  |->  ( N `  ( F `  v )
) ) : A --> S ) )
118, 10mpbird 156 . . . . 5  |-  ( ph  ->  G : A --> S )
1211ffvelrnda 5302 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
134ffvelrnda 5302 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
146ralrimiva 2392 . . . . . . 7  |-  ( ph  ->  A. v  e.  A  ( N `  ( F `
 v ) )  e.  S )
157fnmpt 5025 . . . . . . 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 4989 . . . . . 6  |-  ( ph  ->  ( G  Fn  A  <->  ( v  e.  A  |->  ( N `  ( F `
 v ) ) )  Fn  A ) )
1816, 17mpbird 156 . . . . 5  |-  ( ph  ->  G  Fn  A )
19 dffn5im 5219 . . . . 5  |-  ( G  Fn  A  ->  G  =  ( w  e.  A  |->  ( G `  w ) ) )
2018, 19syl 14 . . . 4  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
214feqmptd 5226 . . . 4  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
221, 12, 13, 20, 21offval2 5726 . . 3  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
239fveq1d 5180 . . . . . . . 8  |-  ( ph  ->  ( G `  w
)  =  ( ( v  e.  A  |->  ( N `  ( F `
 v ) ) ) `  w ) )
2423adantr 261 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( ( v  e.  A  |->  ( N `
 ( F `  v ) ) ) `
 w ) )
25 simpr 103 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  w  e.  A )
262adantr 261 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  A )  ->  N : S --> S )
2726, 13ffvelrnd 5303 . . . . . . . 8  |-  ( (
ph  /\  w  e.  A )  ->  ( N `  ( F `  w ) )  e.  S )
28 fveq2 5178 . . . . . . . . . 10  |-  ( v  =  w  ->  ( F `  v )  =  ( F `  w ) )
2928fveq2d 5182 . . . . . . . . 9  |-  ( v  =  w  ->  ( N `  ( F `  v ) )  =  ( N `  ( F `  w )
) )
3029, 7fvmptg 5248 . . . . . . . 8  |-  ( ( w  e.  A  /\  ( N `  ( F `
 w ) )  e.  S )  -> 
( ( v  e.  A  |->  ( N `  ( F `  v ) ) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3125, 27, 30syl2anc 391 . . . . . . 7  |-  ( (
ph  /\  w  e.  A )  ->  (
( v  e.  A  |->  ( N `  ( F `  v )
) ) `  w
)  =  ( N `
 ( F `  w ) ) )
3224, 31eqtrd 2072 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  =  ( N `  ( F `  w ) ) )
3332oveq1d 5527 . . . . 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 2392 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
3635adantr 261 . . . . . 6  |-  ( (
ph  /\  w  e.  A )  ->  A. x  e.  S  ( ( N `  x ) R x )  =  B )
37 fveq2 5178 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  ( N `  x )  =  ( N `  ( F `  w ) ) )
38 id 19 . . . . . . . . 9  |-  ( x  =  ( F `  w )  ->  x  =  ( F `  w ) )
3937, 38oveq12d 5530 . . . . . . . 8  |-  ( x  =  ( F `  w )  ->  (
( N `  x
) R x )  =  ( ( N `
 ( F `  w ) ) R ( F `  w
) ) )
4039eqeq1d 2048 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
( ( N `  x ) R x )  =  B  <->  ( ( N `  ( F `  w ) ) R ( F `  w
) )  =  B ) )
4140rspcva 2654 . . . . . 6  |-  ( ( ( F `  w
)  e.  S  /\  A. x  e.  S  ( ( N `  x
) R x )  =  B )  -> 
( ( N `  ( F `  w ) ) R ( F `
 w ) )  =  B )
4213, 36, 41syl2anc 391 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  (
( N `  ( F `  w )
) R ( F `
 w ) )  =  B )
4333, 42eqtrd 2072 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( G `  w
) R ( F `
 w ) )  =  B )
4443mpteq2dva 3847 . . 3  |-  ( ph  ->  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) )  =  ( w  e.  A  |->  B ) )
4522, 44eqtrd 2072 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  B ) )
46 fconstmpt 4387 . 2  |-  ( A  X.  { B }
)  =  ( w  e.  A  |->  B )
4745, 46syl6eqr 2090 1  |-  ( ph  ->  ( G  oF R F )  =  ( A  X.  { B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   A.wral 2306   {csn 3375    |-> cmpt 3818    X. cxp 4343    Fn wfn 4897   -->wf 4898   ` cfv 4902  (class class class)co 5512    oFcof 5710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712
This theorem is referenced by: (None)
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