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Theorem fmptcof 6072
Description: Version of fmptco 6071 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 3411 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ R
32nfel1 2596 . . . . . 6  |-  F/ x [_ z  /  x ]_ R  e.  B
4 csbeq1a 3404 . . . . . . 7  |-  ( x  =  z  ->  R  =  [_ z  /  x ]_ R )
54eleq1d 2491 . . . . . 6  |-  ( x  =  z  ->  ( R  e.  B  <->  [_ z  /  x ]_ R  e.  B
) )
63, 5rspc 3176 . . . . 5  |-  ( z  e.  A  ->  ( A. x  e.  A  R  e.  B  ->  [_ z  /  x ]_ R  e.  B )
)
71, 6mpan9 471 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ R  e.  B )
8 fmptcof.2 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
9 nfcv 2580 . . . . . 6  |-  F/_ z R
109, 2, 4cbvmpt 4515 . . . . 5  |-  ( x  e.  A  |->  R )  =  ( z  e.  A  |->  [_ z  /  x ]_ R )
118, 10syl6eq 2479 . . . 4  |-  ( ph  ->  F  =  ( z  e.  A  |->  [_ z  /  x ]_ R ) )
12 fmptcof.3 . . . . 5  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
13 nfcv 2580 . . . . . 6  |-  F/_ w S
14 nfcsb1v 3411 . . . . . 6  |-  F/_ y [_ w  /  y ]_ S
15 csbeq1a 3404 . . . . . 6  |-  ( y  =  w  ->  S  =  [_ w  /  y ]_ S )
1613, 14, 15cbvmpt 4515 . . . . 5  |-  ( y  e.  B  |->  S )  =  ( w  e.  B  |->  [_ w  /  y ]_ S )
1712, 16syl6eq 2479 . . . 4  |-  ( ph  ->  G  =  ( w  e.  B  |->  [_ w  /  y ]_ S
) )
18 csbeq1 3398 . . . 4  |-  ( w  =  [_ z  /  x ]_ R  ->  [_ w  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
197, 11, 17, 18fmptco 6071 . . 3  |-  ( ph  ->  ( G  o.  F
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S ) )
20 nfcv 2580 . . . 4  |-  F/_ z [_ R  /  y ]_ S
21 nfcv 2580 . . . . 5  |-  F/_ x S
222, 21nfcsb 3413 . . . 4  |-  F/_ x [_ [_ z  /  x ]_ R  /  y ]_ S
234csbeq1d 3402 . . . 4  |-  ( x  =  z  ->  [_ R  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
2420, 22, 23cbvmpt 4515 . . 3  |-  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S )
2519, 24syl6eqr 2481 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
26 eqid 2422 . . . 4  |-  A  =  A
27 nfcvd 2581 . . . . . 6  |-  ( R  e.  B  ->  F/_ y T )
28 fmptcof.4 . . . . . 6  |-  ( y  =  R  ->  S  =  T )
2927, 28csbiegf 3419 . . . . 5  |-  ( R  e.  B  ->  [_ R  /  y ]_ S  =  T )
3029ralimi 2815 . . . 4  |-  ( A. x  e.  A  R  e.  B  ->  A. x  e.  A  [_ R  / 
y ]_ S  =  T )
31 mpteq12 4503 . . . 4  |-  ( ( A  =  A  /\  A. x  e.  A  [_ R  /  y ]_ S  =  T )  ->  (
x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
3226, 30, 31sylancr 667 . . 3  |-  ( A. x  e.  A  R  e.  B  ->  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
331, 32syl 17 . 2  |-  ( ph  ->  ( x  e.  A  |-> 
[_ R  /  y ]_ S )  =  ( x  e.  A  |->  T ) )
3425, 33eqtrd 2463 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   A.wral 2771   [_csb 3395    |-> cmpt 4482    o. ccom 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609
This theorem is referenced by:  fmptcos  6073  yonedalem3b  16163  gsumcom2  17606  evl1sca  18921  cnmptk1  20694  cnmpt1k  20695  cnmptkk  20696  cncfmpt1f  21943  copco  22047  pcoass  22053  sincn  23397  coscn  23398  lgseisenlem3  24277  fcomptf  28262  eulerpartgbij  29213  cncfcompt2  37717
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