MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmptcof Structured version   Unicode version

Theorem fmptcof 5895
Description: Version of fmptco 5894 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 3276 . . . . . . 7  |-  F/_ x [_ z  /  x ]_ R
32nfel1 2582 . . . . . 6  |-  F/ x [_ z  /  x ]_ R  e.  B
4 csbeq1a 3252 . . . . . . 7  |-  ( x  =  z  ->  R  =  [_ z  /  x ]_ R )
54eleq1d 2502 . . . . . 6  |-  ( x  =  z  ->  ( R  e.  B  <->  [_ z  /  x ]_ R  e.  B
) )
63, 5rspc 3039 . . . . 5  |-  ( z  e.  A  ->  ( A. x  e.  A  R  e.  B  ->  [_ z  /  x ]_ R  e.  B )
)
71, 6mpan9 456 . . . 4  |-  ( (
ph  /\  z  e.  A )  ->  [_ z  /  x ]_ R  e.  B )
8 fmptcof.2 . . . . 5  |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )
9 nfcv 2572 . . . . . 6  |-  F/_ z R
109, 2, 4cbvmpt 4292 . . . . 5  |-  ( x  e.  A  |->  R )  =  ( z  e.  A  |->  [_ z  /  x ]_ R )
118, 10syl6eq 2484 . . . 4  |-  ( ph  ->  F  =  ( z  e.  A  |->  [_ z  /  x ]_ R ) )
12 fmptcof.3 . . . . 5  |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )
13 nfcv 2572 . . . . . 6  |-  F/_ w S
14 nfcsb1v 3276 . . . . . 6  |-  F/_ y [_ w  /  y ]_ S
15 csbeq1a 3252 . . . . . 6  |-  ( y  =  w  ->  S  =  [_ w  /  y ]_ S )
1613, 14, 15cbvmpt 4292 . . . . 5  |-  ( y  e.  B  |->  S )  =  ( w  e.  B  |->  [_ w  /  y ]_ S )
1712, 16syl6eq 2484 . . . 4  |-  ( ph  ->  G  =  ( w  e.  B  |->  [_ w  /  y ]_ S
) )
18 csbeq1 3247 . . . 4  |-  ( w  =  [_ z  /  x ]_ R  ->  [_ w  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
197, 11, 17, 18fmptco 5894 . . 3  |-  ( ph  ->  ( G  o.  F
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S ) )
20 nfcv 2572 . . . 4  |-  F/_ z [_ R  /  y ]_ S
21 nfcv 2572 . . . . 5  |-  F/_ x S
222, 21nfcsb 3278 . . . 4  |-  F/_ x [_ [_ z  /  x ]_ R  /  y ]_ S
234csbeq1d 3250 . . . 4  |-  ( x  =  z  ->  [_ R  /  y ]_ S  =  [_ [_ z  /  x ]_ R  /  y ]_ S )
2420, 22, 23cbvmpt 4292 . . 3  |-  ( x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( z  e.  A  |->  [_ [_ z  /  x ]_ R  / 
y ]_ S )
2519, 24syl6eqr 2486 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  [_ R  /  y ]_ S
) )
26 eqid 2436 . . . 4  |-  A  =  A
27 nfcvd 2573 . . . . . 6  |-  ( R  e.  B  ->  F/_ y T )
28 fmptcof.4 . . . . . 6  |-  ( y  =  R  ->  S  =  T )
2927, 28csbiegf 3284 . . . . 5  |-  ( R  e.  B  ->  [_ R  /  y ]_ S  =  T )
3029ralimi 2774 . . . 4  |-  ( A. x  e.  A  R  e.  B  ->  A. x  e.  A  [_ R  / 
y ]_ S  =  T )
31 mpteq12 4281 . . . 4  |-  ( ( A  =  A  /\  A. x  e.  A  [_ R  /  y ]_ S  =  T )  ->  (
x  e.  A  |->  [_ R  /  y ]_ S
)  =  ( x  e.  A  |->  T ) )
3226, 30, 31sylancr 645 . . 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 2468 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A  |->  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2698   [_csb 3244    e. cmpt 4259    o. ccom 4875
This theorem is referenced by:  fmptcos  5896  yonedalem3b  14369  gsumcom2  15542  cnmptk1  17706  cnmpt1k  17707  cnmptkk  17708  cncfmpt1f  18936  copco  19036  pcoass  19042  evl1sca  19943  sincn  20353  coscn  20354  lgseisenlem3  21128  fcomptf  24070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455
  Copyright terms: Public domain W3C validator