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Theorem fmptcof 6066
Description: Version of fmptco 6065 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 3456 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ R
32nfel1 2645 . . . . . 6  |-  F/ x [_ z  /  x ]_ R  e.  B
4 csbeq1a 3449 . . . . . . 7  |-  ( x  =  z  ->  R  =  [_ z  /  x ]_ R )
54eleq1d 2536 . . . . . 6  |-  ( x  =  z  ->  ( R  e.  B  <->  [_ z  /  x ]_ R  e.  B
) )
63, 5rspc 3213 . . . . 5  |-  ( z  e.  A  ->  ( A. x  e.  A  R  e.  B  ->  [_ z  /  x ]_ R  e.  B )
)
71, 6mpan9 469 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ R  e.  B )
8 fmptcof.2 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
9 nfcv 2629 . . . . . 6  |-  F/_ z R
109, 2, 4cbvmpt 4543 . . . . 5  |-  ( x  e.  A  |->  R )  =  ( z  e.  A  |->  [_ z  /  x ]_ R )
118, 10syl6eq 2524 . . . 4  |-  ( ph  ->  F  =  ( z  e.  A  |->  [_ z  /  x ]_ R ) )
12 fmptcof.3 . . . . 5  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
13 nfcv 2629 . . . . . 6  |-  F/_ w S
14 nfcsb1v 3456 . . . . . 6  |-  F/_ y [_ w  /  y ]_ S
15 csbeq1a 3449 . . . . . 6  |-  ( y  =  w  ->  S  =  [_ w  /  y ]_ S )
1613, 14, 15cbvmpt 4543 . . . . 5  |-  ( y  e.  B  |->  S )  =  ( w  e.  B  |->  [_ w  /  y ]_ S )
1712, 16syl6eq 2524 . . . 4  |-  ( ph  ->  G  =  ( w  e.  B  |->  [_ w  /  y ]_ S
) )
18 csbeq1 3443 . . . 4  |-  ( w  =  [_ z  /  x ]_ R  ->  [_ w  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
197, 11, 17, 18fmptco 6065 . . 3  |-  ( ph  ->  ( G  o.  F
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S ) )
20 nfcv 2629 . . . 4  |-  F/_ z [_ R  /  y ]_ S
21 nfcv 2629 . . . . 5  |-  F/_ x S
222, 21nfcsb 3458 . . . 4  |-  F/_ x [_ [_ z  /  x ]_ R  /  y ]_ S
234csbeq1d 3447 . . . 4  |-  ( x  =  z  ->  [_ R  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
2420, 22, 23cbvmpt 4543 . . 3  |-  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S )
2519, 24syl6eqr 2526 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
26 eqid 2467 . . . 4  |-  A  =  A
27 nfcvd 2630 . . . . . 6  |-  ( R  e.  B  ->  F/_ y T )
28 fmptcof.4 . . . . . 6  |-  ( y  =  R  ->  S  =  T )
2927, 28csbiegf 3464 . . . . 5  |-  ( R  e.  B  ->  [_ R  /  y ]_ S  =  T )
3029ralimi 2860 . . . 4  |-  ( A. x  e.  A  R  e.  B  ->  A. x  e.  A  [_ R  / 
y ]_ S  =  T )
31 mpteq12 4532 . . . 4  |-  ( ( A  =  A  /\  A. x  e.  A  [_ R  /  y ]_ S  =  T )  ->  (
x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
3226, 30, 31sylancr 663 . . 3  |-  ( A. x  e.  A  R  e.  B  ->  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
331, 32syl 16 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
[_ R  /  y ]_ S )  =  ( x  e.  A  |->  T ) )
3425, 33eqtrd 2508 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817   [_csb 3440    |-> cmpt 4511    o. ccom 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  fmptcos  6067  yonedalem3b  15423  gsumcom2  16876  evl1sca  18240  cnmptk1  20050  cnmpt1k  20051  cnmptkk  20052  cncfmpt1f  21285  copco  21386  pcoass  21392  sincn  22706  coscn  22707  lgseisenlem3  23492  fcomptf  27326  cncfcompt  31544
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