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Theorem fmptcof 5704
Description: Version of fmptco 5703 where  ph needn't be distinct from  x. (Contributed by NM, 27-Dec-2014.)
Hypotheses
Ref Expression
fmptcof.1  |-  ( ph  ->  A. x  e.  A  R  e.  B )
fmptcof.2  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
fmptcof.3  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
fmptcof.4  |-  ( y  =  R  ->  S  =  T )
Assertion
Ref Expression
fmptcof  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Distinct variable groups:    x, y, B   
y, R    x, S    x, A    y, T
Allowed substitution hints:    ph( x, y)    A( y)    R( x)    S( y)    T( x)    F( x, y)    G( x, y)

Proof of Theorem fmptcof
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmptcof.1 . . . . 5  |-  ( ph  ->  A. x  e.  A  R  e.  B )
2 nfcsb1v 3105 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ R
32nfel1 2343 . . . . . 6  |-  F/ x [_ z  /  x ]_ R  e.  B
4 csbeq1a 3081 . . . . . . 7  |-  ( x  =  z  ->  R  =  [_ z  /  x ]_ R )
54eleq1d 2258 . . . . . 6  |-  ( x  =  z  ->  ( R  e.  B  <->  [_ z  /  x ]_ R  e.  B
) )
63, 5rspc 2850 . . . . 5  |-  ( z  e.  A  ->  ( A. x  e.  A  R  e.  B  ->  [_ z  /  x ]_ R  e.  B )
)
71, 6mpan9 281 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ R  e.  B )
8 fmptcof.2 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
9 nfcv 2332 . . . . . 6  |-  F/_ z R
109, 2, 4cbvmpt 4113 . . . . 5  |-  ( x  e.  A  |->  R )  =  ( z  e.  A  |->  [_ z  /  x ]_ R )
118, 10eqtrdi 2238 . . . 4  |-  ( ph  ->  F  =  ( z  e.  A  |->  [_ z  /  x ]_ R ) )
12 fmptcof.3 . . . . 5  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
13 nfcv 2332 . . . . . 6  |-  F/_ w S
14 nfcsb1v 3105 . . . . . 6  |-  F/_ y [_ w  /  y ]_ S
15 csbeq1a 3081 . . . . . 6  |-  ( y  =  w  ->  S  =  [_ w  /  y ]_ S )
1613, 14, 15cbvmpt 4113 . . . . 5  |-  ( y  e.  B  |->  S )  =  ( w  e.  B  |->  [_ w  /  y ]_ S )
1712, 16eqtrdi 2238 . . . 4  |-  ( ph  ->  G  =  ( w  e.  B  |->  [_ w  /  y ]_ S
) )
18 csbeq1 3075 . . . 4  |-  ( w  =  [_ z  /  x ]_ R  ->  [_ w  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
197, 11, 17, 18fmptco 5703 . . 3  |-  ( ph  ->  ( G  o.  F
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S ) )
20 nfcv 2332 . . . 4  |-  F/_ z [_ R  /  y ]_ S
21 nfcv 2332 . . . . 5  |-  F/_ x S
222, 21nfcsb 3109 . . . 4  |-  F/_ x [_ [_ z  /  x ]_ R  /  y ]_ S
234csbeq1d 3079 . . . 4  |-  ( x  =  z  ->  [_ R  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
2420, 22, 23cbvmpt 4113 . . 3  |-  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S )
2519, 24eqtr4di 2240 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
26 eqid 2189 . . . 4  |-  A  =  A
27 nfcvd 2333 . . . . . 6  |-  ( R  e.  B  ->  F/_ y T )
28 fmptcof.4 . . . . . 6  |-  ( y  =  R  ->  S  =  T )
2927, 28csbiegf 3115 . . . . 5  |-  ( R  e.  B  ->  [_ R  /  y ]_ S  =  T )
3029ralimi 2553 . . . 4  |-  ( A. x  e.  A  R  e.  B  ->  A. x  e.  A  [_ R  / 
y ]_ S  =  T )
31 mpteq12 4101 . . . 4  |-  ( ( A  =  A  /\  A. x  e.  A  [_ R  /  y ]_ S  =  T )  ->  (
x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
3226, 30, 31sylancr 414 . . 3  |-  ( A. x  e.  A  R  e.  B  ->  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
331, 32syl 14 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
[_ R  /  y ]_ S )  =  ( x  e.  A  |->  T ) )
3425, 33eqtrd 2222 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   A.wral 2468   [_csb 3072    |-> cmpt 4079    o. ccom 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by:  fmptcos  5705  cncfmpt1f  14544  sincn  14650  coscn  14651
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