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Theorem offval22 6877
Description: The function operation expressed as a mapping, variation of offval2 6553. (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 6598 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  e.  _V )
41, 2, 3syl2anc 665 . . 3  |-  ( ph  ->  ( A  X.  B
)  e.  _V )
5 xp1st 6828 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  ( 1st `  z )  e.  A )
6 xp2nd 6829 . . . . 5  |-  ( z  e.  ( A  X.  B )  ->  ( 2nd `  z )  e.  B )
75, 6jca 534 . . . 4  |-  ( z  e.  ( A  X.  B )  ->  (
( 1st `  z
)  e.  A  /\  ( 2nd `  z )  e.  B ) )
8 fvex 5882 . . . . . 6  |-  ( 2nd `  z )  e.  _V
9 fvex 5882 . . . . . 6  |-  ( 1st `  z )  e.  _V
10 nfcv 2582 . . . . . . 7  |-  F/_ y
( 2nd `  z
)
11 nfcv 2582 . . . . . . 7  |-  F/_ x
( 2nd `  z
)
12 nfcv 2582 . . . . . . 7  |-  F/_ x
( 1st `  z
)
13 nfv 1751 . . . . . . . 8  |-  F/ y ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )
14 nfcsb1v 3408 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ C
1514nfel1 2598 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ C  e.  _V
1613, 15nfim 1975 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
17 nfv 1751 . . . . . . . 8  |-  F/ x
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
)
18 nfcsb1v 3408 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C
1918nfel1 2598 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V
2017, 19nfim 1975 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
21 eleq1 2492 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  ( y  e.  B  <->  ( 2nd `  z
)  e.  B ) )
22213anbi3d 1341 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( ( ph  /\  x  e.  A  /\  y  e.  B
)  <->  ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B ) ) )
23 csbeq1a 3401 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  C  =  [_ ( 2nd `  z
)  /  y ]_ C )
2423eleq1d 2489 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( C  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
2522, 24imbi12d 321 . . . . . . 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 2492 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  ( x  e.  A  <->  ( 1st `  z
)  e.  A ) )
27263anbi2d 1340 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( ( ph  /\  x  e.  A  /\  ( 2nd `  z
)  e.  B )  <-> 
( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) ) )
28 csbeq1a 3401 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ C  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C )
2928eleq1d 2489 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ C  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
)
3027, 29imbi12d 321 . . . . . . 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 3087 . . . . . . . 8  |-  ( C  e.  X  ->  C  e.  _V )
3331, 32syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  C  e.  _V )
3410, 11, 12, 16, 20, 25, 30, 33vtocl2gf 3138 . . . . . 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 676 . . . . 5  |-  ( (
ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C  e.  _V )
36353expb 1206 . . . 4  |-  ( (
ph  /\  ( ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
377, 36sylan2 476 . . 3  |-  ( (
ph  /\  z  e.  ( A  X.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ C  e.  _V )
38 nfcsb1v 3408 . . . . . . . . 9  |-  F/_ y [_ ( 2nd `  z
)  /  y ]_ D
3938nfel1 2598 . . . . . . . 8  |-  F/ y
[_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4013, 39nfim 1975 . . . . . . 7  |-  F/ y ( ( ph  /\  x  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
41 nfcsb1v 3408 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D
4241nfel1 2598 . . . . . . . 8  |-  F/ x [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V
4317, 42nfim 1975 . . . . . . 7  |-  F/ x
( ( ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
44 csbeq1a 3401 . . . . . . . . 9  |-  ( y  =  ( 2nd `  z
)  ->  D  =  [_ ( 2nd `  z
)  /  y ]_ D )
4544eleq1d 2489 . . . . . . . 8  |-  ( y  =  ( 2nd `  z
)  ->  ( D  e.  _V  <->  [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
)
4622, 45imbi12d 321 . . . . . . 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 3401 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  [_ ( 2nd `  z )  /  y ]_ D  =  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D )
4847eleq1d 2489 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( [_ ( 2nd `  z )  /  y ]_ D  e.  _V  <->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
)
4927, 48imbi12d 321 . . . . . . 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 3087 . . . . . . . 8  |-  ( D  e.  Y  ->  D  e.  _V )
5250, 51syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  A  /\  y  e.  B
)  ->  D  e.  _V )
5310, 11, 12, 40, 43, 46, 49, 52vtocl2gf 3138 . . . . . 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 676 . . . . 5  |-  ( (
ph  /\  ( 1st `  z )  e.  A  /\  ( 2nd `  z
)  e.  B )  ->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D  e.  _V )
55543expb 1206 . . . 4  |-  ( (
ph  /\  ( ( 1st `  z )  e.  A  /\  ( 2nd `  z )  e.  B
) )  ->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D  e.  _V )
567, 55sylan2 476 . . 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 6862 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ C )
5957, 58syl6eq 2477 . . 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 6862 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  D )  =  ( z  e.  ( A  X.  B
)  |->  [_ ( 1st `  z
)  /  x ]_ [_ ( 2nd `  z
)  /  y ]_ D )
6260, 61syl6eq 2477 . . 3  |-  ( ph  ->  G  =  ( z  e.  ( A  X.  B )  |->  [_ ( 1st `  z )  /  x ]_ [_ ( 2nd `  z )  /  y ]_ D ) )
634, 37, 56, 59, 62offval2 6553 . 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 6332 . . . . . . 7  |-  ( ( 2nd `  z )  e.  _V  ->  [_ ( 2nd `  z )  / 
y ]_ ( C R D )  =  (
[_ ( 2nd `  z
)  /  y ]_ C R [_ ( 2nd `  z )  /  y ]_ D ) )
6564csbeq2dv 3806 . . . . . 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 6332 . . . . . 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 2450 . . . 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 4500 . . 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 6862 . . 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 2452 . 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 2477 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078   [_csb 3392    |-> cmpt 4475    X. cxp 4843   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298    oFcof 6534   1stc1st 6796   2ndc2nd 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6536  df-1st 6798  df-2nd 6799
This theorem is referenced by:  matsc  19399  mdetrsca2  19553  mdetrlin2  19556  mdetunilem5  19565  smadiadetglem2  19621  mat2pmatghm  19678  pm2mpghm  19764
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