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Theorem offval22 6862
Description: The function operation expressed as a mapping, variation of offval2 6540. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
offval22.a  |-  ( ph  ->  A  e.  V )
offval22.b  |-  ( ph  ->  B  e.  W )
offval22.c  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  e.  X )
offval22.d  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  D  e.  Y )
offval22.f  |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  C ) )
offval22.g  |-  ( ph  ->  G  =  ( x  e.  A ,  y  e.  B  |->  D ) )
Assertion
Ref Expression
offval22  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A ,  y  e.  B  |->  ( C R D ) ) )
Distinct variable groups:    ph, x, y   
x, A, y    x, B, y    x, R, y
Allowed substitution hints:    C( x, y)    D( x, y)    F( x, y)    G( x, y)    V( x, y)    W( x, y)    X( x, y)    Y( x, y)

Proof of Theorem offval22
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 offval22.a . . . 4  |-  ( ph  ->  A  e.  V )
2 offval22.b . . . 4  |-  ( ph  ->  B  e.  W )
3 xpexg 6586 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
5 xp1st 6814 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  ( 1st `  z )  e.  A )
6 xp2nd 6815 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  ( 2nd `  z )  e.  B )
75, 6jca 532 . . . 4  |-  ( z  e.  ( A  X.  B )  ->  (
( 1st `  z
)  e.  A  /\  ( 2nd `  z )  e.  B ) )
8 fvex 5876 . . . . . 6  |-  ( 2nd `  z )  e.  _V
9 fvex 5876 . . . . . 6  |-  ( 1st `  z )  e.  _V
10 nfcv 2629 . . . . . . 7  |-  F/_ y
( 2nd `  z
)
11 nfcv 2629 . . . . . . 7  |-  F/_ x
( 2nd `  z
)
12 nfcv 2629 . . . . . . 7  |-  F/_ x
( 1st `  z
)
13 nfv 1683 . . . . . . . 8  |-  F/ y ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )
14 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
1514nfel1 2645 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
1613, 15nfim 1867 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
17 nfv 1683 . . . . . . . 8  |-  F/ x
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
)
18 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1918nfel1 2645 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
2017, 19nfim 1867 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
21 eleq1 2539 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( y  e.  B  <->  ( 2nd `  z
)  e.  B ) )
22213anbi3d 1305 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( ( ph  /\  x  e.  A  /\  y  e.  B
)  <->  ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B ) ) )
23 csbeq1a 3444 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
2423eleq1d 2536 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2522, 24imbi12d 320 . . . . . . 7  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( ph  /\  x  e.  A  /\  y  e.  B )  ->  C  e.  _V )  <->  ( ( ph  /\  x  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
) )
26 eleq1 2539 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  e.  A  <->  ( 1st `  z
)  e.  A ) )
27263anbi2d 1304 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( ( ph  /\  x  e.  A  /\  ( 2nd `  z
)  e.  B )  <-> 
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) ) )
28 csbeq1a 3444 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2928eleq1d 2536 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
3027, 29imbi12d 320 . . . . . . 7  |-  ( x  =  ( 1st `  z
)  ->  ( (
( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z )  / 
y ]_ C  e.  _V ) 
<->  ( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V ) ) )
31 offval22.c . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  e.  X )
32 elex 3122 . . . . . . . 8  |-  ( C  e.  X  ->  C  e.  _V )
3331, 32syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  e.  _V )
3410, 11, 12, 16, 20, 25, 30, 33vtocl2gf 3173 . . . . . 6  |-  ( ( ( 2nd `  z
)  e.  _V  /\  ( 1st `  z )  e.  _V )  -> 
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V ) )
358, 9, 34mp2an 672 . . . . 5  |-  ( (
ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
36353expb 1197 . . . 4  |-  ( (
ph  /\  ( ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
377, 36sylan2 474 . . 3  |-  ( (
ph  /\  z  e.  ( A  X.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
38 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ D
3938nfel1 2645 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4013, 39nfim 1867 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
41 nfcsb1v 3451 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D
4241nfel1 2645 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4317, 42nfim 1867 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
44 csbeq1a 3444 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  D  =  [_ ( 2nd `  z
)  /  y ]_ D )
4544eleq1d 2536 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( D  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
)
4622, 45imbi12d 320 . . . . . . 7  |-  ( y  =  ( 2nd `  z
)  ->  ( (
( ph  /\  x  e.  A  /\  y  e.  B )  ->  D  e.  _V )  <->  ( ( ph  /\  x  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
) )
47 csbeq1a 3444 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ D  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D )
4847eleq1d 2536 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ D  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
)
4927, 48imbi12d 320 . . . . . . 7  |-  ( x  =  ( 1st `  z
)  ->  ( (
( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z )  / 
y ]_ D  e.  _V ) 
<->  ( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V ) ) )
50 offval22.d . . . . . . . 8  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  D  e.  Y )
51 elex 3122 . . . . . . . 8  |-  ( D  e.  Y  ->  D  e.  _V )
5250, 51syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  D  e.  _V )
5310, 11, 12, 40, 43, 46, 49, 52vtocl2gf 3173 . . . . . 6  |-  ( ( ( 2nd `  z
)  e.  _V  /\  ( 1st `  z )  e.  _V )  -> 
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V ) )
548, 9, 53mp2an 672 . . . . 5  |-  ( (
ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
55543expb 1197 . . . 4  |-  ( (
ph  /\  ( ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
567, 55sylan2 474 . . 3  |-  ( (
ph  /\  z  e.  ( A  X.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
57 offval22.f . . . 4  |-  ( ph  ->  F  =  ( x  e.  A ,  y  e.  B  |->  C ) )
58 mpt2mpts 6848 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
5957, 58syl6eq 2524 . . 3  |-  ( ph  ->  F  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C ) )
60 offval22.g . . . 4  |-  ( ph  ->  G  =  ( x  e.  A ,  y  e.  B  |->  D ) )
61 mpt2mpts 6848 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D )
6260, 61syl6eq 2524 . . 3  |-  ( ph  ->  G  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D ) )
634, 37, 56, 59, 62offval2 6540 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( z  e.  ( A  X.  B ) 
|->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D ) ) )
64 csbov12g 6318 . . . . . . 7  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 2nd `  z )  / 
y ]_ ( C R D )  =  (
[_ ( 2nd `  z
)  /  y ]_ C R [_ ( 2nd `  z )  /  y ]_ D ) )
6564csbeq2dv 3835 . . . . . 6  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ ( C R D )  =  [_ ( 1st `  z )  /  x ]_ ( [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 2nd `  z )  /  y ]_ D
) )
668, 65ax-mp 5 . . . . 5  |-  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ ( C R D )  =  [_ ( 1st `  z )  /  x ]_ ( [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 2nd `  z )  /  y ]_ D
)
67 csbov12g 6318 . . . . . 6  |-  ( ( 1st `  z )  e.  _V  ->  [_ ( 1st `  z )  /  x ]_ ( [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 2nd `  z )  /  y ]_ D
)  =  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ D ) )
689, 67ax-mp 5 . . . . 5  |-  [_ ( 1st `  z )  /  x ]_ ( [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 2nd `  z )  /  y ]_ D
)  =  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ D )
6966, 68eqtr2i 2497 . . . 4  |-  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ D )  = 
[_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ ( C R D )
7069mpteq2i 4530 . . 3  |-  ( z  e.  ( A  X.  B )  |->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ D ) )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ ( C R D ) )
71 mpt2mpts 6848 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  ( C R D ) )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ ( C R D ) )
7270, 71eqtr4i 2499 . 2  |-  ( z  e.  ( A  X.  B )  |->  ( [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ C R [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  / 
y ]_ D ) )  =  ( x  e.  A ,  y  e.  B  |->  ( C R D ) )
7363, 72syl6eq 2524 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  A ,  y  e.  B  |->  ( C R D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435    |-> cmpt 4505    X. cxp 4997   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286    oFcof 6522   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-1st 6784  df-2nd 6785
This theorem is referenced by:  matsc  18747  mdetrsca2  18901  mdetrlin2  18904  mdetunilem5  18913  smadiadetglem2  18969  mat2pmatghm  19026  pm2mpghm  19112
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