Theorem fprodcnv 11617

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.

Step	Hyp	Ref	Expression
1		csbeq1a 3066	. . . 4 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
2		2ndexg 6163	. . . . . 6 |- ( y e. _V -> ( 2nd ` y ) e. _V )
3	2	elv 2741	. . . . 5 |- ( 2nd ` y ) e. _V
4		1stexg 6162	. . . . . 6 |- ( y e. _V -> ( 1st ` y ) e. _V )
5	4	elv 2741	. . . . 5 |- ( 1st ` y ) e. _V
6		vex 2740	. . . . . . . 8 |- j e. _V
7		vex 2740	. . . . . . . 8 |- k e. _V
8	6, 7	opex 4226	. . . . . . 7 |- <. j , k >. e. _V
9		fprodcnv.1	. . . . . . 7 |- ( x = <. j , k >. -> B = D )
10	8, 9	csbie 3102	. . . . . 6 |- [_ <. j , k >. / x ]_ B = D
11		opeq12 3778	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. )
12	11	csbeq1d 3064	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
13	10, 12	eqtr3id 2224	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
14	3, 5, 13	csbie2 3106	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B
15	1, 14	eqtr4di 2228	. . 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 6934	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
19	16, 17, 18	syl2anc 411	. . 3 |- ( ph -> `' A e. Fin )
20		relcnv 5002	. . . . 5 |- Rel `' A
21		cnvf1o 6220	. . . . 5 |- ( Rel `' A -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A )
22	20, 21	ax-mp 5	. . . 4 |- ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A
23		dfrel2 5075	. . . . . 6 |- ( Rel A <-> `' `' A = A )
24	16, 23	sylib 122	. . . . 5 |- ( ph -> `' `' A = A )
25	24	f1oeq3d 5454	. . . 4 |- ( ph -> ( ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> `' `' A <-> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A ) )
26	22, 25	mpbii 148	. . 3 |- ( ph -> ( z e. `' A |-> U. `' { z } ) : `' A -1-1-onto-> A )
27		1st2nd 6176	. . . . . . 7 |- ( ( Rel `' A /\ y e. `' A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
28	20, 27	mpan 424	. . . . . 6 |- ( y e. `' A -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
29	28	fveq2d 5515	. . . . 5 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
30	28	eleq1d 2246	. . . . . . 7 |- ( y e. `' A -> ( y e. `' A <-> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A ) )
31	30	ibi 176	. . . . . 6 |- ( y e. `' A -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A )
32		sneq 3602	. . . . . . . . . 10 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
33	32	cnveqd 4799	. . . . . . . . 9 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
34	33	unieqd 3818	. . . . . . . 8 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
35		opswapg 5111	. . . . . . . . 9 |- ( ( ( 1st ` y ) e. _V /\ ( 2nd ` y ) e. _V ) -> U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
36	5, 3, 35	mp2an 426	. . . . . . . 8 |- U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >.
37	34, 36	eqtrdi 2226	. . . . . . 7 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
38		eqid 2177	. . . . . . 7 |- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } )
39	3, 5	opex 4226	. . . . . . 7 |- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V
40	37, 38, 39	fvmpt 5589	. . . . . 6 |- ( <. ( 1st ` y ) , ( 2nd ` y ) >. e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
41	31, 40	syl 14	. . . . 5 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
42	29, 41	eqtrd 2210	. . . 4 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = <. ( 2nd ` y ) , ( 1st ` y ) >. )
43	42	adantl 277	. . 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 )
45	15, 19, 26, 43, 44	fprodf1o 11580	. 2 |- ( ph -> prod_ x e. A B = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
46		csbeq1a 3066	. . . . 5 |- ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
47	28, 46	syl 14	. . . 4 |- ( y e. `' A -> C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
48	7, 6	opex 4226	. . . . . . 7 |- <. k , j >. e. _V
49		fprodcnv.2	. . . . . . 7 |- ( y = <. k , j >. -> C = D )
50	48, 49	csbie 3102	. . . . . 6 |- [_ <. k , j >. / y ]_ C = D
51		opeq12 3778	. . . . . . . 8 |- ( ( k = ( 1st ` y ) /\ j = ( 2nd ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
52	51	ancoms 268	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. k , j >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
53	52	csbeq1d 3064	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
54	50, 53	eqtr3id 2224	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
55	3, 5, 54	csbie2 3106	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C
56	47, 55	eqtr4di 2228	. . 3 |- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
57	56	prodeq2i 11554	. 2 |- prod_ y e. `' A C = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D
58	45, 57	eqtr4di 2228	1 |- ( ph -> prod_ x e. A B = prod_ y e. `' A C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   [_csb 3057   {csn 3591   <.cop 3594   U.cuni 3807   |-> cmpt 4061   `'ccnv 4622   Rel wrel 4628   -1-1-onto->wf1o 5211   ` cfv 5212   1stc1st 6133   2ndc2nd 6134   Fincfn 6734   CCcc 7800   prod_cprod 11542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543

This theorem is referenced by:  fprodcom2fi  11618