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Theorem fmptcof 5874
Description: Version of fmptco 5873 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 3301 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ R
32nfel1 2587 . . . . . 6  |-  F/ x [_ z  /  x ]_ R  e.  B
4 csbeq1a 3294 . . . . . . 7  |-  ( x  =  z  ->  R  =  [_ z  /  x ]_ R )
54eleq1d 2507 . . . . . 6  |-  ( x  =  z  ->  ( R  e.  B  <->  [_ z  /  x ]_ R  e.  B
) )
63, 5rspc 3064 . . . . 5  |-  ( z  e.  A  ->  ( A. x  e.  A  R  e.  B  ->  [_ z  /  x ]_ R  e.  B )
)
71, 6mpan9 466 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ R  e.  B )
8 fmptcof.2 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
9 nfcv 2577 . . . . . 6  |-  F/_ z R
109, 2, 4cbvmpt 4379 . . . . 5  |-  ( x  e.  A  |->  R )  =  ( z  e.  A  |->  [_ z  /  x ]_ R )
118, 10syl6eq 2489 . . . 4  |-  ( ph  ->  F  =  ( z  e.  A  |->  [_ z  /  x ]_ R ) )
12 fmptcof.3 . . . . 5  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
13 nfcv 2577 . . . . . 6  |-  F/_ w S
14 nfcsb1v 3301 . . . . . 6  |-  F/_ y [_ w  /  y ]_ S
15 csbeq1a 3294 . . . . . 6  |-  ( y  =  w  ->  S  =  [_ w  /  y ]_ S )
1613, 14, 15cbvmpt 4379 . . . . 5  |-  ( y  e.  B  |->  S )  =  ( w  e.  B  |->  [_ w  /  y ]_ S )
1712, 16syl6eq 2489 . . . 4  |-  ( ph  ->  G  =  ( w  e.  B  |->  [_ w  /  y ]_ S
) )
18 csbeq1 3288 . . . 4  |-  ( w  =  [_ z  /  x ]_ R  ->  [_ w  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
197, 11, 17, 18fmptco 5873 . . 3  |-  ( ph  ->  ( G  o.  F
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S ) )
20 nfcv 2577 . . . 4  |-  F/_ z [_ R  /  y ]_ S
21 nfcv 2577 . . . . 5  |-  F/_ x S
222, 21nfcsb 3303 . . . 4  |-  F/_ x [_ [_ z  /  x ]_ R  /  y ]_ S
234csbeq1d 3292 . . . 4  |-  ( x  =  z  ->  [_ R  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
2420, 22, 23cbvmpt 4379 . . 3  |-  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S )
2519, 24syl6eqr 2491 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
26 eqid 2441 . . . 4  |-  A  =  A
27 nfcvd 2578 . . . . . 6  |-  ( R  e.  B  ->  F/_ y T )
28 fmptcof.4 . . . . . 6  |-  ( y  =  R  ->  S  =  T )
2927, 28csbiegf 3309 . . . . 5  |-  ( R  e.  B  ->  [_ R  /  y ]_ S  =  T )
3029ralimi 2789 . . . 4  |-  ( A. x  e.  A  R  e.  B  ->  A. x  e.  A  [_ R  / 
y ]_ S  =  T )
31 mpteq12 4368 . . . 4  |-  ( ( A  =  A  /\  A. x  e.  A  [_ R  /  y ]_ S  =  T )  ->  (
x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
3226, 30, 31sylancr 658 . . 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 2473 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   A.wral 2713   [_csb 3285    e. cmpt 4347    o. ccom 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423
This theorem is referenced by:  fmptcos  5875  yonedalem3b  15085  gsumcom2  16457  evl1sca  17737  cnmptk1  19213  cnmpt1k  19214  cnmptkk  19215  cncfmpt1f  20448  copco  20549  pcoass  20555  sincn  21868  coscn  21869  lgseisenlem3  22649  fcomptf  25915
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