Theorem fprodcnv 12392

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 3156	. . . 4 |- ( x = <. ( 2nd ` y ) , ( 1st ` y ) >. -> B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
2		2ndexg 6402	. . . . . 6 |- ( y e. _V -> ( 2nd ` y ) e. _V )
3	2	elv 2825	. . . . 5 |- ( 2nd ` y ) e. _V
4		1stexg 6401	. . . . . 6 |- ( y e. _V -> ( 1st ` y ) e. _V )
5	4	elv 2825	. . . . 5 |- ( 1st ` y ) e. _V
6		vex 2824	. . . . . . . 8 |- j e. _V
7		vex 2824	. . . . . . . 8 |- k e. _V
8	6, 7	opex 4369	. . . . . . 7 |- <. j , k >. e. _V
9		fprodcnv.1	. . . . . . 7 |- ( x = <. j , k >. -> B = D )
10	8, 9	csbie 3193	. . . . . 6 |- [_ <. j , k >. / x ]_ B = D
11		opeq12 3906	. . . . . . 7 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> <. j , k >. = <. ( 2nd ` y ) , ( 1st ` y ) >. )
12	11	csbeq1d 3154	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. j , k >. / x ]_ B = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
13	10, 12	eqtr3id 2285	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B )
14	3, 5, 13	csbie2 3197	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 2nd ` y ) , ( 1st ` y ) >. / x ]_ B
15	1, 14	eqtr4di 2289	. . 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 7255	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
19	16, 17, 18	syl2anc 415	. . 3 |- ( ph -> `' A e. Fin )
20		relcnv 5165	. . . . 5 |- Rel `' A
21		cnvf1o 6461	. . . . 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 5238	. . . . . 6 |- ( Rel A <-> `' `' A = A )
24	16, 23	sylib 122	. . . . 5 |- ( ph -> `' `' A = A )
25	24	f1oeq3d 5636	. . . 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 6415	. . . . . . 7 |- ( ( Rel `' A /\ y e. `' A ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
28	20, 27	mpan 428	. . . . . 6 |- ( y e. `' A -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
29	28	fveq2d 5699	. . . . 5 |- ( y e. `' A -> ( ( z e. `' A |-> U. `' { z } ) ` y ) = ( ( z e. `' A |-> U. `' { z } ) ` <. ( 1st ` y ) , ( 2nd ` y ) >. ) )
30	28	eleq1d 2307	. . . . . . 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 3720	. . . . . . . . . 10 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> { z } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
33	32	cnveqd 4956	. . . . . . . . 9 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> `' { z } = `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
34	33	unieqd 3946	. . . . . . . 8 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
35		opswapg 5274	. . . . . . . . 9 |- ( ( ( 1st ` y ) e. _V /\ ( 2nd ` y ) e. _V ) -> U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
36	5, 3, 35	mp2an 430	. . . . . . . 8 |- U. `' { <. ( 1st ` y ) , ( 2nd ` y ) >. } = <. ( 2nd ` y ) , ( 1st ` y ) >.
37	34, 36	eqtrdi 2287	. . . . . . 7 |- ( z = <. ( 1st ` y ) , ( 2nd ` y ) >. -> U. `' { z } = <. ( 2nd ` y ) , ( 1st ` y ) >. )
38		eqid 2238	. . . . . . 7 |- ( z e. `' A |-> U. `' { z } ) = ( z e. `' A |-> U. `' { z } )
39	3, 5	opex 4369	. . . . . . 7 |- <. ( 2nd ` y ) , ( 1st ` y ) >. e. _V
40	37, 38, 39	fvmpt 5782	. . . . . 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 2271	. . . 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 12355	. 2 |- ( ph -> prod_ x e. A B = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
46		csbeq1a 3156	. . . . 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 4369	. . . . . . 7 |- <. k , j >. e. _V
49		fprodcnv.2	. . . . . . 7 |- ( y = <. k , j >. -> C = D )
50	48, 49	csbie 3193	. . . . . 6 |- [_ <. k , j >. / y ]_ C = D
51		opeq12 3906	. . . . . . . 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 3154	. . . . . 6 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> [_ <. k , j >. / y ]_ C = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
54	50, 53	eqtr3id 2285	. . . . 5 |- ( ( j = ( 2nd ` y ) /\ k = ( 1st ` y ) ) -> D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C )
55	3, 5, 54	csbie2 3197	. . . 4 |- [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D = [_ <. ( 1st ` y ) , ( 2nd ` y ) >. / y ]_ C
56	47, 55	eqtr4di 2289	. . 3 |- ( y e. `' A -> C = [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D )
57	56	prodeq2i 12329	. 2 |- prod_ y e. `' A C = prod_ y e. `' A [_ ( 2nd ` y ) / j ]_ [_ ( 1st ` y ) / k ]_ D
58	45, 57	eqtr4di 2289	1 |- ( ph -> prod_ x e. A B = prod_ y e. `' A C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   [_csb 3147   {csn 3709   <.cop 3712   U.cuni 3935   |-> cmpt 4192   `'ccnv 4773   Rel wrel 4779   -1-1-onto->wf1o 5376   ` cfv 5377   1stc1st 6372   2ndc2nd 6373   Fincfn 7022   CCcc 8177   prod_cprod 12317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-proddc 12318

This theorem is used by:  fprodcom2fi  12393