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Theorem fprodcnv 27423
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 3294 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 fvex 5698 . . . . 5  |-  ( 2nd `  y )  e.  _V
3 fvex 5698 . . . . 5  |-  ( 1st `  y )  e.  _V
4 opex 4553 . . . . . . 7  |-  <. j ,  k >.  e.  _V
5 nfcv 2577 . . . . . . 7  |-  F/_ x D
6 fprodcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
74, 5, 6csbief 3310 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
8 opeq12 4058 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
98csbeq1d 3292 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
107, 9syl5eqr 2487 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
112, 3, 10csbie2 3314 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
121, 11syl6eqr 2491 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
13 fprodcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
14 cnvfi 7591 . . . 4  |-  ( A  e.  Fin  ->  `' A  e.  Fin )
1513, 14syl 16 . . 3  |-  ( ph  ->  `' A  e.  Fin )
16 relcnv 5203 . . . . 5  |-  Rel  `' A
17 cnvf1o 6670 . . . . 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 5285 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2119, 20sylib 196 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
22 f1oeq3 5631 . . . . 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 6619 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2616, 25mpan 665 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2726fveq2d 5692 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
2826eleq1d 2507 . . . . . . 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 3884 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3130cnveqd 5011 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3231unieqd 4098 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
33 opswap 5323 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3432, 33syl6eq 2489 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
35 eqid 2441 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
36 opex 4553 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
3734, 35, 36fvmpt 5771 . . . . . 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 2473 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4039adantl 463 . . 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 27388 . 2  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
43 csbeq1a 3294 . . . . 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 4553 . . . . . . 7  |-  <. k ,  j >.  e.  _V
46 nfcv 2577 . . . . . . 7  |-  F/_ y D
47 fprodcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
4845, 46, 47csbief 3310 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
49 opeq12 4058 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5049ancoms 450 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5150csbeq1d 3292 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5248, 51syl5eqr 2487 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
532, 3, 52csbie2 3314 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5444, 53syl6eqr 2491 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5554prodeq2i 27361 . 2  |-  prod_ y  e.  `'  A C  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y )  / 
k ]_ D
5642, 55syl6eqr 2491 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 1364    e. wcel 1761   [_csb 3285   {csn 3874   <.cop 3880   U.cuni 4088    e. cmpt 4347   `'ccnv 4835   Rel wrel 4841   -1-1-onto->wf1o 5414   ` cfv 5415   1stc1st 6574   2ndc2nd 6575   Fincfn 7306   CCcc 9276   prod_cprod 27347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-prod 27348
This theorem is referenced by:  fprodcom2  27424
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