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Theorem offval22 6657
Description: The function operation expressed as a mapping, variation of offval2 6341. (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 6512 . . . 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 6611 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  ( 1st `  z )  e.  A )
6 xp2nd 6612 . . . . 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 5706 . . . . . 6  |-  ( 2nd `  z )  e.  _V
9 fvex 5706 . . . . . 6  |-  ( 1st `  z )  e.  _V
10 nfcv 2584 . . . . . . 7  |-  F/_ y
( 2nd `  z
)
11 nfcv 2584 . . . . . . 7  |-  F/_ x
( 2nd `  z
)
12 nfcv 2584 . . . . . . 7  |-  F/_ x
( 1st `  z
)
13 nfv 1673 . . . . . . . 8  |-  F/ y ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )
14 nfcsb1v 3309 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
1514nfel1 2594 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
1613, 15nfim 1853 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
17 nfv 1673 . . . . . . . 8  |-  F/ x
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
)
18 nfcsb1v 3309 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1918nfel1 2594 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
2017, 19nfim 1853 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
21 eleq1 2503 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( y  e.  B  <->  ( 2nd `  z
)  e.  B ) )
22213anbi3d 1295 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( ( ph  /\  x  e.  A  /\  y  e.  B
)  <->  ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B ) ) )
23 csbeq1a 3302 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
2423eleq1d 2509 . . . . . . . 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 2503 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  e.  A  <->  ( 1st `  z
)  e.  A ) )
27263anbi2d 1294 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( ( ph  /\  x  e.  A  /\  ( 2nd `  z
)  e.  B )  <-> 
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) ) )
28 csbeq1a 3302 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2928eleq1d 2509 . . . . . . . 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 2986 . . . . . . . 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 3037 . . . . . 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 1188 . . . 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 3309 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ D
3938nfel1 2594 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4013, 39nfim 1853 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
41 nfcsb1v 3309 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D
4241nfel1 2594 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4317, 42nfim 1853 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
44 csbeq1a 3302 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  D  =  [_ ( 2nd `  z
)  /  y ]_ D )
4544eleq1d 2509 . . . . . . . 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 3302 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ D  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D )
4847eleq1d 2509 . . . . . . . 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 2986 . . . . . . . 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 3037 . . . . . 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 1188 . . . 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 6643 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
5957, 58syl6eq 2491 . . 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 6643 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D )
6260, 61syl6eq 2491 . . 3  |-  ( ph  ->  G  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D ) )
634, 37, 56, 59, 62offval2 6341 . 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 6130 . . . . . . 7  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 2nd `  z )  / 
y ]_ ( C R D )  =  (
[_ ( 2nd `  z
)  /  y ]_ C R [_ ( 2nd `  z )  /  y ]_ D ) )
6564csbeq2dv 3692 . . . . . 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 6130 . . . . . 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 2464 . . . 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 4380 . . 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 6643 . . 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 2466 . 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 2491 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   [_csb 3293    e. cmpt 4355    X. cxp 4843   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323   1stc1st 6580   2ndc2nd 6581
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-1st 6582  df-2nd 6583
This theorem is referenced by:  matsc  18346  mdetrsca2  18416  mdetrlin2  18418  mdetunilem5  18427  smadiadetglem2  18483  pmattomply1ghm  30930
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