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Theorem fmpt2co 6659
Description: Composition of two functions. Variation of fmptco 5879 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 2811 . . . . 5  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  R  e.  C )
3 eqid 2443 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( x  e.  A ,  y  e.  B  |->  R )
43fmpt2 6644 . . . . 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 2582 . . . . . . 7  |-  F/_ u R
7 nfcv 2582 . . . . . . 7  |-  F/_ v R
8 nfcv 2582 . . . . . . . 8  |-  F/_ x
v
9 nfcsb1v 3307 . . . . . . . 8  |-  F/_ x [_ u  /  x ]_ R
108, 9nfcsb 3309 . . . . . . 7  |-  F/_ x [_ v  /  y ]_ [_ u  /  x ]_ R
11 nfcsb1v 3307 . . . . . . 7  |-  F/_ y [_ v  /  y ]_ [_ u  /  x ]_ R
12 csbeq1a 3300 . . . . . . . 8  |-  ( x  =  u  ->  R  =  [_ u  /  x ]_ R )
13 csbeq1a 3300 . . . . . . . 8  |-  ( y  =  v  ->  [_ u  /  x ]_ R  = 
[_ v  /  y ]_ [_ u  /  x ]_ R )
1412, 13sylan9eq 2495 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  v )  ->  R  =  [_ v  /  y ]_ [_ u  /  x ]_ R )
156, 7, 10, 11, 14cbvmpt2 6168 . . . . . 6  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( u  e.  A ,  v  e.  B  |->  [_ v  /  y ]_ [_ u  /  x ]_ R )
16 vex 2978 . . . . . . . . . 10  |-  u  e. 
_V
17 vex 2978 . . . . . . . . . 10  |-  v  e. 
_V
1816, 17op2ndd 6591 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  ( 2nd `  w
)  =  v )
1918csbeq1d 3298 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R )
2016, 17op1std 6590 . . . . . . . . . 10  |-  ( w  =  <. u ,  v
>.  ->  ( 1st `  w
)  =  u )
2120csbeq1d 3298 . . . . . . . . 9  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 1st `  w
)  /  x ]_ R  =  [_ u  /  x ]_ R )
2221csbeq2dv 3690 . . . . . . . 8  |-  ( w  =  <. u ,  v
>.  ->  [_ v  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2319, 22eqtrd 2475 . . . . . . 7  |-  ( w  =  <. u ,  v
>.  ->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  =  [_ v  / 
y ]_ [_ u  /  x ]_ R )
2423mpt2mpt 6185 . . . . . 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 2466 . . . . 5  |-  ( x  e.  A ,  y  e.  B  |->  R )  =  ( w  e.  ( A  X.  B
)  |->  [_ ( 2nd `  w
)  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R )
2625fmpt 5867 . . . 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 2491 . . 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 5881 . 2  |-  ( ph  ->  ( G  o.  F
)  =  ( w  e.  ( A  X.  B )  |->  [_ [_ ( 2nd `  w )  / 
y ]_ [_ ( 1st `  w )  /  x ]_ R  /  z ]_ S ) )
3223csbeq1d 3298 . . . . 5  |-  ( w  =  <. u ,  v
>.  ->  [_ [_ ( 2nd `  w )  /  y ]_ [_ ( 1st `  w
)  /  x ]_ R  /  z ]_ S  =  [_ [_ v  / 
y ]_ [_ u  /  x ]_ R  /  z ]_ S )
3332mpt2mpt 6185 . . . 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 2582 . . . . 5  |-  F/_ u [_ R  /  z ]_ S
35 nfcv 2582 . . . . 5  |-  F/_ v [_ R  /  z ]_ S
36 nfcv 2582 . . . . . 6  |-  F/_ x S
3710, 36nfcsb 3309 . . . . 5  |-  F/_ x [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
38 nfcv 2582 . . . . . 6  |-  F/_ y S
3911, 38nfcsb 3309 . . . . 5  |-  F/_ y [_ [_ v  /  y ]_ [_ u  /  x ]_ R  /  z ]_ S
4014csbeq1d 3298 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  [_ R  /  z ]_ S  =  [_ [_ v  /  y ]_ [_ u  /  x ]_ R  / 
z ]_ S )
4134, 35, 37, 39, 40cbvmpt2 6168 . . . 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 2466 . . 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 1183 . . . . 5  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  R  e.  C )
44 nfcvd 2583 . . . . . 6  |-  ( R  e.  C  ->  F/_ z T )
45 fmpt2co.4 . . . . . 6  |-  ( z  =  R  ->  S  =  T )
4644, 45csbiegf 3315 . . . . 5  |-  ( R  e.  C  ->  [_ R  /  z ]_ S  =  T )
4743, 46syl 16 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  [_ R  / 
z ]_ S  =  T )
4847mpt2eq3dva 6153 . . 3  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |-> 
[_ R  /  z ]_ S )  =  ( x  e.  A , 
y  e.  B  |->  T ) )
4942, 48syl5eq 2487 . 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 2475 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 965    = wceq 1369    e. wcel 1756   A.wral 2718   [_csb 3291   <.cop 3886    e. cmpt 4353    X. cxp 4841    o. ccom 4847   -->wf 5417   ` cfv 5421    e. cmpt2 6096   1stc1st 6578   2ndc2nd 6579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-oprab 6098  df-mpt2 6099  df-1st 6580  df-2nd 6581
This theorem is referenced by:  oprabco  6660  evlslem2  17600  txswaphmeolem  19380  xpstopnlem1  19385  stdbdxmet  20093  rrxds  20900  cnre2csqima  26344  cvmlift2lem7  27201
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