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Theorem fmpt2co 6202
Description: Composition of two functions. Variation of fmptco 5691 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 2635 . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  R  e.  C )
3 eqid 2283 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( x  e.  A ,  y  e.  B  |->  R )
43fmpt2 6191 . . . . 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 188 . . . 4  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  R ) : ( A  X.  B ) --> C )
6 nfcv 2419 . . . . . . 7  |-  F/_ u R
7 nfcv 2419 . . . . . . 7  |-  F/_ v R
8 nfcv 2419 . . . . . . . 8  |-  F/_ x
v
9 nfcsb1v 3113 . . . . . . . 8  |-  F/_ x [_ u  /  x ]_ R
108, 9nfcsb 3115 . . . . . . 7  |-  F/_ x [_ v  /  y ]_ [_ u  /  x ]_ R
11 nfcsb1v 3113 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ [_ u  /  x ]_ R
12 csbeq1a 3089 . . . . . . . 8  |-  ( x  =  u  ->  R  =  [_ u  /  x ]_ R )
13 csbeq1a 3089 . . . . . . . 8  |-  ( y  =  v  ->  [_ u  /  x ]_ R  = 
[_ v  /  y ]_ [_ u  /  x ]_ R )
1412, 13sylan9eq 2335 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  v )  ->  R  =  [_ v  /  y ]_ [_ u  /  x ]_ R )
156, 7, 10, 11, 14cbvmpt2 5925 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( u  e.  A ,  v  e.  B  |->  [_ v  /  y ]_ [_ u  /  x ]_ R )
16 vex 2791 . . . . . . . . . 10  |-  u  e. 
_V
17 vex 2791 . . . . . . . . . 10  |-  v  e. 
_V
1816, 17op2ndd 6131 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  ( 2nd `  w
)  =  v )
1918csbeq1d 3087 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R )
2016, 17op1std 6130 . . . . . . . . . 10  |-  ( w  =  <. u ,  v
>.  ->  ( 1st `  w
)  =  u )
2120csbeq1d 3087 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 1st `  w
)  /  x ]_ R  =  [_ u  /  x ]_ R )
2221csbeq2dv 3106 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ v  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2319, 22eqtrd 2315 . . . . . . 7  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2423mpt2mpt 5939 . . . . . 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 2306 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R )
2625fmpt 5681 . . . 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 203 . . 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 2331 . . 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 5693 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S ) )
3223csbeq1d 3087 . . . . 5  |-  ( w  =  <. u ,  v
>.  ->  [_ [_ ( 2nd `  w )  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  /  z ]_ S  =  [_ [_ v  / 
y ]_ [_ u  /  x ]_ R  /  z ]_ S )
3332mpt2mpt 5939 . . . 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 2419 . . . . 5  |-  F/_ u [_ R  /  z ]_ S
35 nfcv 2419 . . . . 5  |-  F/_ v [_ R  /  z ]_ S
36 nfcv 2419 . . . . . 6  |-  F/_ x S
3710, 36nfcsb 3115 . . . . 5  |-  F/_ x [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
38 nfcv 2419 . . . . . 6  |-  F/_ y S
3911, 38nfcsb 3115 . . . . 5  |-  F/_ y [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
4014csbeq1d 3087 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  [_ R  /  z ]_ S  =  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  / 
z ]_ S )
4134, 35, 37, 39, 40cbvmpt2 5925 . . . 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 2306 . . 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 1147 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  R  e.  C )
44 nfcvd 2420 . . . . . 6  |-  ( R  e.  C  ->  F/_ z T )
45 fmpt2co.4 . . . . . 6  |-  ( z  =  R  ->  S  =  T )
4644, 45csbiegf 3121 . . . . 5  |-  ( R  e.  C  ->  [_ R  /  z ]_ S  =  T )
4743, 46syl 15 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  [_ R  / 
z ]_ S  =  T )
4847mpt2eq3dva 5912 . . 3  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |-> 
[_ R  /  z ]_ S )  =  ( x  e.  A , 
y  e.  B  |->  T ) )
4942, 48syl5eq 2327 . 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 2315 1  |-  ( ph  ->  ( G  o.  F
)  =  ( x  e.  A ,  y  e.  B  |->  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081   <.cop 3643    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  oprabco  6203  evlslem2  16249  txswaphmeolem  17495  xpstopnlem1  17500  stdbdxmet  18061  cnre2csqima  23295  cvmlift2lem7  23840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123
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