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Theorem fprodcnv 11526
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 3040 . . . 4  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
2 2ndexg 6117 . . . . . 6  |-  ( y  e.  _V  ->  ( 2nd `  y )  e. 
_V )
32elv 2716 . . . . 5  |-  ( 2nd `  y )  e.  _V
4 1stexg 6116 . . . . . 6  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
54elv 2716 . . . . 5  |-  ( 1st `  y )  e.  _V
6 vex 2715 . . . . . . . 8  |-  j  e. 
_V
7 vex 2715 . . . . . . . 8  |-  k  e. 
_V
86, 7opex 4190 . . . . . . 7  |-  <. j ,  k >.  e.  _V
9 fprodcnv.1 . . . . . . 7  |-  ( x  =  <. j ,  k
>.  ->  B  =  D )
108, 9csbie 3076 . . . . . 6  |-  [_ <. j ,  k >.  /  x ]_ B  =  D
11 opeq12 3744 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. j ,  k >.  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
1211csbeq1d 3038 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. j ,  k >.  /  x ]_ B  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B )
1310, 12eqtr3id 2204 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 2nd `  y
) ,  ( 1st `  y ) >.  /  x ]_ B )
143, 5, 13csbie2 3080 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 2nd `  y ) ,  ( 1st `  y
) >.  /  x ]_ B
151, 14eqtr4di 2208 . . 3  |-  ( x  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.  ->  B  =  [_ ( 2nd `  y
)  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
16 fprodcnv.4 . . . 4  |-  ( ph  ->  Rel  A )
17 fprodcnv.3 . . . 4  |-  ( ph  ->  A  e.  Fin )
18 relcnvfi 6886 . . . 4  |-  ( ( Rel  A  /\  A  e.  Fin )  ->  `' A  e.  Fin )
1916, 17, 18syl2anc 409 . . 3  |-  ( ph  ->  `' A  e.  Fin )
20 relcnv 4965 . . . . 5  |-  Rel  `' A
21 cnvf1o 6173 . . . . 5  |-  ( Rel  `' A  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A )
2220, 21ax-mp 5 . . . 4  |-  ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A
23 dfrel2 5037 . . . . . 6  |-  ( Rel 
A  <->  `' `' A  =  A
)
2416, 23sylib 121 . . . . 5  |-  ( ph  ->  `' `' A  =  A
)
2524f1oeq3d 5412 . . . 4  |-  ( ph  ->  ( ( z  e.  `' A  |->  U. `' { z } ) : `' A -1-1-onto-> `' `' A 
<->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A ) )
2622, 25mpbii 147 . . 3  |-  ( ph  ->  ( z  e.  `' A  |->  U. `' { z } ) : `' A
-1-1-onto-> A )
27 1st2nd 6130 . . . . . . 7  |-  ( ( Rel  `' A  /\  y  e.  `' A
)  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2820, 27mpan 421 . . . . . 6  |-  ( y  e.  `' A  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2928fveq2d 5473 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  =  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
)
3028eleq1d 2226 . . . . . . 7  |-  ( y  e.  `' A  -> 
( y  e.  `' A 
<-> 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A ) )
3130ibi 175 . . . . . 6  |-  ( y  e.  `' A  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A )
32 sneq 3571 . . . . . . . . . 10  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  { z }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3332cnveqd 4763 . . . . . . . . 9  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  `' { z }  =  `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
3433unieqd 3784 . . . . . . . 8  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
35 opswapg 5073 . . . . . . . . 9  |-  ( ( ( 1st `  y
)  e.  _V  /\  ( 2nd `  y )  e.  _V )  ->  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
365, 3, 35mp2an 423 . . . . . . . 8  |-  U. `' { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >.
3734, 36eqtrdi 2206 . . . . . . 7  |-  ( z  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  U. `' { z }  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
38 eqid 2157 . . . . . . 7  |-  ( z  e.  `' A  |->  U. `' { z } )  =  ( z  e.  `' A  |->  U. `' { z } )
393, 5opex 4190 . . . . . . 7  |-  <. ( 2nd `  y ) ,  ( 1st `  y
) >.  e.  _V
4037, 38, 39fvmpt 5546 . . . . . 6  |-  ( <.
( 1st `  y
) ,  ( 2nd `  y ) >.  e.  `' A  ->  ( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4131, 40syl 14 . . . . 5  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 <. ( 1st `  y
) ,  ( 2nd `  y ) >. )  =  <. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4229, 41eqtrd 2190 . . . 4  |-  ( y  e.  `' A  -> 
( ( z  e.  `' A  |->  U. `' { z } ) `
 y )  = 
<. ( 2nd `  y
) ,  ( 1st `  y ) >. )
4342adantl 275 . . 3  |-  ( (
ph  /\  y  e.  `' A )  ->  (
( z  e.  `' A  |->  U. `' { z } ) `  y
)  =  <. ( 2nd `  y ) ,  ( 1st `  y
) >. )
44 fprodcnv.5 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4515, 19, 26, 43, 44fprodf1o 11489 . 2  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
46 csbeq1a 3040 . . . . 5  |-  ( y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  ->  C  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
4728, 46syl 14 . . . 4  |-  ( y  e.  `' A  ->  C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
487, 6opex 4190 . . . . . . 7  |-  <. k ,  j >.  e.  _V
49 fprodcnv.2 . . . . . . 7  |-  ( y  =  <. k ,  j
>.  ->  C  =  D )
5048, 49csbie 3076 . . . . . 6  |-  [_ <. k ,  j >.  /  y ]_ C  =  D
51 opeq12 3744 . . . . . . . 8  |-  ( ( k  =  ( 1st `  y )  /\  j  =  ( 2nd `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5251ancoms 266 . . . . . . 7  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  <. k ,  j >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
5352csbeq1d 3038 . . . . . 6  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  [_ <. k ,  j >.  /  y ]_ C  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C )
5450, 53eqtr3id 2204 . . . . 5  |-  ( ( j  =  ( 2nd `  y )  /\  k  =  ( 1st `  y
) )  ->  D  =  [_ <. ( 1st `  y
) ,  ( 2nd `  y ) >.  /  y ]_ C )
553, 5, 54csbie2 3080 . . . 4  |-  [_ ( 2nd `  y )  / 
j ]_ [_ ( 1st `  y )  /  k ]_ D  =  [_ <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /  y ]_ C
5647, 55eqtr4di 2208 . . 3  |-  ( y  e.  `' A  ->  C  =  [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y
)  /  k ]_ D )
5756prodeq2i 11463 . 2  |-  prod_ y  e.  `'  A C  =  prod_ y  e.  `'  A [_ ( 2nd `  y )  /  j ]_ [_ ( 1st `  y )  / 
k ]_ D
5845, 57eqtr4di 2208 1  |-  ( ph  ->  prod_ x  e.  A  B  =  prod_ y  e.  `'  A C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   [_csb 3031   {csn 3560   <.cop 3563   U.cuni 3773    |-> cmpt 4026   `'ccnv 4586   Rel wrel 4592   -1-1-onto->wf1o 5170   ` cfv 5171   1stc1st 6087   2ndc2nd 6088   Fincfn 6686   CCcc 7731   prod_cprod 11451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-frec 6339  df-1o 6364  df-oadd 6368  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-fz 9914  df-fzo 10046  df-seqfrec 10349  df-exp 10423  df-ihash 10654  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-clim 11180  df-proddc 11452
This theorem is referenced by:  fprodcom2fi  11527
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