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Theorem fmpt2co 6856
Description: Composition of two functions. Variation of fmptco 6045 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
fmpt2co.1  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  R  e.  C )
fmpt2co.2  |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  R ) )
fmpt2co.3  |-  ( ph  ->  G  =  ( z  e.  C  |->  S ) )
fmpt2co.4  |-  ( z  =  R  ->  S  =  T )
Assertion
Ref Expression
fmpt2co  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A ,  y  e.  B  |->  T ) )
Distinct variable groups:    x, y, B    x, z, C, y    ph, x, y    x, S, y    x, A, y   
z, R    z, T
Allowed substitution hints:    ph( z)    A( z)    B( z)    R( x, y)    S( z)    T( x, y)    F( x, y, z)    G( x, y, z)

Proof of Theorem fmpt2co
Dummy variables  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmpt2co.1 . . . . . 6  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  R  e.  C )
21ralrimivva 2878 . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  R  e.  C )
3 eqid 2460 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( x  e.  A ,  y  e.  B  |->  R )
43fmpt2 6841 . . . . 5  |-  ( A. x  e.  A  A. y  e.  B  R  e.  C  <->  ( x  e.  A ,  y  e.  B  |->  R ) : ( A  X.  B
) --> C )
52, 4sylib 196 . . . 4  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  R ) : ( A  X.  B ) --> C )
6 nfcv 2622 . . . . . . 7  |-  F/_ u R
7 nfcv 2622 . . . . . . 7  |-  F/_ v R
8 nfcv 2622 . . . . . . . 8  |-  F/_ x
v
9 nfcsb1v 3444 . . . . . . . 8  |-  F/_ x [_ u  /  x ]_ R
108, 9nfcsb 3446 . . . . . . 7  |-  F/_ x [_ v  /  y ]_ [_ u  /  x ]_ R
11 nfcsb1v 3444 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ [_ u  /  x ]_ R
12 csbeq1a 3437 . . . . . . . 8  |-  ( x  =  u  ->  R  =  [_ u  /  x ]_ R )
13 csbeq1a 3437 . . . . . . . 8  |-  ( y  =  v  ->  [_ u  /  x ]_ R  = 
[_ v  /  y ]_ [_ u  /  x ]_ R )
1412, 13sylan9eq 2521 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  v )  ->  R  =  [_ v  /  y ]_ [_ u  /  x ]_ R )
156, 7, 10, 11, 14cbvmpt2 6351 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( u  e.  A ,  v  e.  B  |->  [_ v  /  y ]_ [_ u  /  x ]_ R )
16 vex 3109 . . . . . . . . . 10  |-  u  e. 
_V
17 vex 3109 . . . . . . . . . 10  |-  v  e. 
_V
1816, 17op2ndd 6785 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  ( 2nd `  w
)  =  v )
1918csbeq1d 3435 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R )
2016, 17op1std 6784 . . . . . . . . . 10  |-  ( w  =  <. u ,  v
>.  ->  ( 1st `  w
)  =  u )
2120csbeq1d 3435 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 1st `  w
)  /  x ]_ R  =  [_ u  /  x ]_ R )
2221csbeq2dv 3828 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ v  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2319, 22eqtrd 2501 . . . . . . 7  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2423mpt2mpt 6369 . . . . . 6  |-  ( w  e.  ( A  X.  B )  |->  [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R )  =  ( u  e.  A , 
v  e.  B  |->  [_ v  /  y ]_ [_ u  /  x ]_ R )
2515, 24eqtr4i 2492 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R )
2625fmpt 6033 . . . 4  |-  ( A. w  e.  ( A  X.  B ) [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  e.  C  <->  ( x  e.  A , 
y  e.  B  |->  R ) : ( A  X.  B ) --> C )
275, 26sylibr 212 . . 3  |-  ( ph  ->  A. w  e.  ( A  X.  B )
[_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  e.  C )
28 fmpt2co.2 . . . 4  |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  R ) )
2928, 25syl6eq 2517 . . 3  |-  ( ph  ->  F  =  ( w  e.  ( A  X.  B )  |->  [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R ) )
30 fmpt2co.3 . . 3  |-  ( ph  ->  G  =  ( z  e.  C  |->  S ) )
3127, 29, 30fmptcos 6047 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S ) )
3223csbeq1d 3435 . . . . 5  |-  ( w  =  <. u ,  v
>.  ->  [_ [_ ( 2nd `  w )  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  /  z ]_ S  =  [_ [_ v  / 
y ]_ [_ u  /  x ]_ R  /  z ]_ S )
3332mpt2mpt 6369 . . . 4  |-  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S )  =  ( u  e.  A , 
v  e.  B  |->  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
)
34 nfcv 2622 . . . . 5  |-  F/_ u [_ R  /  z ]_ S
35 nfcv 2622 . . . . 5  |-  F/_ v [_ R  /  z ]_ S
36 nfcv 2622 . . . . . 6  |-  F/_ x S
3710, 36nfcsb 3446 . . . . 5  |-  F/_ x [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
38 nfcv 2622 . . . . . 6  |-  F/_ y S
3911, 38nfcsb 3446 . . . . 5  |-  F/_ y [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
4014csbeq1d 3435 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  [_ R  /  z ]_ S  =  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  / 
z ]_ S )
4134, 35, 37, 39, 40cbvmpt2 6351 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  [_ R  /  z ]_ S
)  =  ( u  e.  A ,  v  e.  B  |->  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  / 
z ]_ S )
4233, 41eqtr4i 2492 . . 3  |-  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S )  =  ( x  e.  A , 
y  e.  B  |->  [_ R  /  z ]_ S
)
4313impb 1187 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  R  e.  C )
44 nfcvd 2623 . . . . . 6  |-  ( R  e.  C  ->  F/_ z T )
45 fmpt2co.4 . . . . . 6  |-  ( z  =  R  ->  S  =  T )
4644, 45csbiegf 3452 . . . . 5  |-  ( R  e.  C  ->  [_ R  /  z ]_ S  =  T )
4743, 46syl 16 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  [_ R  / 
z ]_ S  =  T )
4847mpt2eq3dva 6336 . . 3  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |-> 
[_ R  /  z ]_ S )  =  ( x  e.  A , 
y  e.  B  |->  T ) )
4942, 48syl5eq 2513 . 2  |-  ( ph  ->  ( w  e.  ( A  X.  B ) 
|->  [_ [_ ( 2nd `  w )  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  /  z ]_ S
)  =  ( x  e.  A ,  y  e.  B  |->  T ) )
5031, 49eqtrd 2501 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A ,  y  e.  B  |->  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   [_csb 3428   <.cop 4026    |-> cmpt 4498    X. cxp 4990    o. ccom 4996   -->wf 5575   ` cfv 5579    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775
This theorem is referenced by:  oprabco  6857  evlslem2  17944  txswaphmeolem  20033  xpstopnlem1  20038  stdbdxmet  20746  rrxds  21553  cnre2csqima  27515  cvmlift2lem7  28380
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