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Theorem fmpt2co 6393
Description: Composition of two functions. Variation of fmptco 5864 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 2762 . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  R  e.  C )
3 eqid 2408 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( x  e.  A ,  y  e.  B  |->  R )
43fmpt2 6381 . . . . 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 189 . . . 4  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  R ) : ( A  X.  B ) --> C )
6 nfcv 2544 . . . . . . 7  |-  F/_ u R
7 nfcv 2544 . . . . . . 7  |-  F/_ v R
8 nfcv 2544 . . . . . . . 8  |-  F/_ x
v
9 nfcsb1v 3247 . . . . . . . 8  |-  F/_ x [_ u  /  x ]_ R
108, 9nfcsb 3249 . . . . . . 7  |-  F/_ x [_ v  /  y ]_ [_ u  /  x ]_ R
11 nfcsb1v 3247 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ [_ u  /  x ]_ R
12 csbeq1a 3223 . . . . . . . 8  |-  ( x  =  u  ->  R  =  [_ u  /  x ]_ R )
13 csbeq1a 3223 . . . . . . . 8  |-  ( y  =  v  ->  [_ u  /  x ]_ R  = 
[_ v  /  y ]_ [_ u  /  x ]_ R )
1412, 13sylan9eq 2460 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  v )  ->  R  =  [_ v  /  y ]_ [_ u  /  x ]_ R )
156, 7, 10, 11, 14cbvmpt2 6114 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( u  e.  A ,  v  e.  B  |->  [_ v  /  y ]_ [_ u  /  x ]_ R )
16 vex 2923 . . . . . . . . . 10  |-  u  e. 
_V
17 vex 2923 . . . . . . . . . 10  |-  v  e. 
_V
1816, 17op2ndd 6321 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  ( 2nd `  w
)  =  v )
1918csbeq1d 3221 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R )
2016, 17op1std 6320 . . . . . . . . . 10  |-  ( w  =  <. u ,  v
>.  ->  ( 1st `  w
)  =  u )
2120csbeq1d 3221 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 1st `  w
)  /  x ]_ R  =  [_ u  /  x ]_ R )
2221csbeq2dv 3240 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ v  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2319, 22eqtrd 2440 . . . . . . 7  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2423mpt2mpt 6128 . . . . . 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 2431 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R )
2625fmpt 5853 . . . 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 204 . . 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 2456 . . 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 5866 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S ) )
3223csbeq1d 3221 . . . . 5  |-  ( w  =  <. u ,  v
>.  ->  [_ [_ ( 2nd `  w )  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  /  z ]_ S  =  [_ [_ v  / 
y ]_ [_ u  /  x ]_ R  /  z ]_ S )
3332mpt2mpt 6128 . . . 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 2544 . . . . 5  |-  F/_ u [_ R  /  z ]_ S
35 nfcv 2544 . . . . 5  |-  F/_ v [_ R  /  z ]_ S
36 nfcv 2544 . . . . . 6  |-  F/_ x S
3710, 36nfcsb 3249 . . . . 5  |-  F/_ x [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
38 nfcv 2544 . . . . . 6  |-  F/_ y S
3911, 38nfcsb 3249 . . . . 5  |-  F/_ y [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
4014csbeq1d 3221 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  [_ R  /  z ]_ S  =  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  / 
z ]_ S )
4134, 35, 37, 39, 40cbvmpt2 6114 . . . 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 2431 . . 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 1149 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  R  e.  C )
44 nfcvd 2545 . . . . . 6  |-  ( R  e.  C  ->  F/_ z T )
45 fmpt2co.4 . . . . . 6  |-  ( z  =  R  ->  S  =  T )
4644, 45csbiegf 3255 . . . . 5  |-  ( R  e.  C  ->  [_ R  /  z ]_ S  =  T )
4743, 46syl 16 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  [_ R  / 
z ]_ S  =  T )
4847mpt2eq3dva 6101 . . 3  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |-> 
[_ R  /  z ]_ S )  =  ( x  e.  A , 
y  e.  B  |->  T ) )
4942, 48syl5eq 2452 . 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 2440 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A ,  y  e.  B  |->  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   [_csb 3215   <.cop 3781    e. cmpt 4230    X. cxp 4839    o. ccom 4845   -->wf 5413   ` cfv 5417    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311
This theorem is referenced by:  oprabco  6394  evlslem2  16527  txswaphmeolem  17793  xpstopnlem1  17798  stdbdxmet  18502  cnre2csqima  24266  cvmlift2lem7  24953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313
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