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Theorem fprodcnv 27494
Description: Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
Hypotheses
Ref Expression
fprodcnv.1  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
fprodcnv.2  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
fprodcnv.3  |-  ( ph  ->  A  e.  Fin )
fprodcnv.4  |-  ( ph  ->  Rel  A )
fprodcnv.5  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
Assertion
Ref Expression
fprodcnv  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A C )
Distinct variable groups:    x, A, y    B, j, k, y    C, j, k    x, D, y    j, k, x, y    ph, x, y
Allowed substitution hints:    ph( j, k)    A( j, k)    B( x)    C( x, y)    D( j, k)

Proof of Theorem fprodcnv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 csbeq1a 3297 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 fvex 5701 . . . . 5  |-  ( 2nd `  y )  e.  _V
3 fvex 5701 . . . . 5  |-  ( 1st `  y )  e.  _V
4 opex 4556 . . . . . . 7  |-  <. j ,  k >.  e.  _V
5 nfcv 2579 . . . . . . 7  |-  F/_ x D
6 fprodcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
74, 5, 6csbief 3313 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
8 opeq12 4061 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
98csbeq1d 3295 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
107, 9syl5eqr 2489 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
112, 3, 10csbie2 3317 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
121, 11syl6eqr 2493 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
13 fprodcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
14 cnvfi 7595 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  Fin )
1513, 14syl 16 . . 3  |-  ( ph  ->  `' A  e.  Fin )
16 relcnv 5206 . . . . 5  |-  Rel  `' A
17 cnvf1o 6671 . . . . 5  |-  ( Rel  `' A  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A )
1816, 17ax-mp 5 . . . 4  |-  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A
19 fprodcnv.4 . . . . . 6  |-  ( ph  ->  Rel  A )
20 dfrel2 5288 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2119, 20sylib 196 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
22 f1oeq3 5634 . . . . 5  |-  ( `' `' A  =  A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2321, 22syl 16 . . . 4  |-  ( ph  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2418, 23mpbii 211 . . 3  |-  ( ph  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A )
25 1st2nd 6620 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2616, 25mpan 670 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2726fveq2d 5695 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
2826eleq1d 2509 . . . . . . 7  |-  ( y  e.  `' A  -> 
( y  e.  `' A 
<-> 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A ) )
2928ibi 241 . . . . . 6  |-  ( y  e.  `' A  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A )
30 sneq 3887 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3130cnveqd 5015 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3231unieqd 4101 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
33 opswap 5326 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3432, 33syl6eq 2491 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
35 eqid 2443 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
36 opex 4556 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
3734, 35, 36fvmpt 5774 . . . . . 6  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3829, 37syl 16 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
3927, 38eqtrd 2475 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4039adantl 466 . . 3  |-  ( (
ph  /\  y  e.  `' A )  ->  (
( z  e.  `' A  |->  U. `' { z } ) `  y
)  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
41 fprodcnv.5 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4212, 15, 24, 40, 41fprodf1o 27459 . 2  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
43 csbeq1a 3297 . . . . 5  |-  ( y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  C  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
4426, 43syl 16 . . . 4  |-  ( y  e.  `' A  ->  C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
45 opex 4556 . . . . . . 7  |-  <. k ,  j >.  e.  _V
46 nfcv 2579 . . . . . . 7  |-  F/_ y D
47 fprodcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
4845, 46, 47csbief 3313 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
49 opeq12 4061 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5049ancoms 453 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5150csbeq1d 3295 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5248, 51syl5eqr 2489 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
532, 3, 52csbie2 3317 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5444, 53syl6eqr 2493 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5554prodeq2i 27432 . 2  |-  prod_ y  e.  `'  A C  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y )  / 
k ]_ D
5642, 55syl6eqr 2493 1  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   [_csb 3288   {csn 3877   <.cop 3883   U.cuni 4091    e. cmpt 4350   `'ccnv 4839   Rel wrel 4845   -1-1-onto->wf1o 5417   ` cfv 5418   1stc1st 6575   2ndc2nd 6576   Fincfn 7310   CCcc 9280   prod_cprod 27418
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-prod 27419
This theorem is referenced by:  fprodcom2  27495
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