Theorem txdis 14456

Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txdis	|- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )

Proof of Theorem txdis

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 14264	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		distop 14264	. . . . 5 |- ( B e. W -> ~P B e. Top )
3		unipw 4247	. . . . . . 7 |- U. ~P A = A
4	3	eqcomi 2197	. . . . . 6 |- A = U. ~P A
5		unipw 4247	. . . . . . 7 |- U. ~P B = B
6	5	eqcomi 2197	. . . . . 6 |- B = U. ~P B
7	4, 6	txuni 14442	. . . . 5 |- ( ( ~P A e. Top /\ ~P B e. Top ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
8	1, 2, 7	syl2an 289	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
9		eqimss2 3235	. . . 4 |- ( ( A X. B ) = U. ( ~P A tX ~P B ) -> U. ( ~P A tX ~P B ) C_ ( A X. B ) )
10	8, 9	syl 14	. . 3 |- ( ( A e. V /\ B e. W ) -> U. ( ~P A tX ~P B ) C_ ( A X. B ) )
11		sspwuni 3998	. . 3 |- ( ( ~P A tX ~P B ) C_ ~P ( A X. B ) <-> U. ( ~P A tX ~P B ) C_ ( A X. B ) )
12	10, 11	sylibr 134	. 2 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) C_ ~P ( A X. B ) )
13		elelpwi 3614	. . . . . . . . 9 |- ( ( y e. x /\ x e. ~P ( A X. B ) ) -> y e. ( A X. B ) )
14	13	adantl 277	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. ( A X. B ) )
15		xp1st 6220	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 1st ` y ) e. A )
16		snelpwi 4242	. . . . . . . 8 |- ( ( 1st ` y ) e. A -> { ( 1st ` y ) } e. ~P A )
17	14, 15, 16	3syl 17	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { ( 1st ` y ) } e. ~P A )
18		xp2nd 6221	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 2nd ` y ) e. B )
19		snelpwi 4242	. . . . . . . 8 |- ( ( 2nd ` y ) e. B -> { ( 2nd ` y ) } e. ~P B )
20	14, 18, 19	3syl 17	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { ( 2nd ` y ) } e. ~P B )
21		vsnid 3651	. . . . . . . 8 |- y e. { y }
22		1st2nd2 6230	. . . . . . . . . 10 |- ( y e. ( A X. B ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
23	14, 22	syl 14	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
24	23	sneqd 3632	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { y } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
25	21, 24	eleqtrid 2282	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
26		simprl 529	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. x )
27	23, 26	eqeltrrd 2271	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> <. ( 1st ` y ) , ( 2nd ` y ) >. e. x )
28	27	snssd 3764	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x )
29		xpeq1 4674	. . . . . . . . . 10 |- ( z = { ( 1st ` y ) } -> ( z X. w ) = ( { ( 1st ` y ) } X. w ) )
30	29	eleq2d 2263	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( y e. ( z X. w ) <-> y e. ( { ( 1st ` y ) } X. w ) ) )
31	29	sseq1d 3209	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( ( z X. w ) C_ x <-> ( { ( 1st ` y ) } X. w ) C_ x ) )
32	30, 31	anbi12d 473	. . . . . . . 8 |- ( z = { ( 1st ` y ) } -> ( ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) <-> ( y e. ( { ( 1st ` y ) } X. w ) /\ ( { ( 1st ` y ) } X. w ) C_ x ) ) )
33		xpeq2 4675	. . . . . . . . . . 11 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) )
34		1stexg 6222	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
35	34	elv 2764	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
36		2ndexg 6223	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 2nd ` y ) e. _V )
37	36	elv 2764	. . . . . . . . . . . 12 |- ( 2nd ` y ) e. _V
38	35, 37	xpsn 5735	. . . . . . . . . . 11 |- ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. }
39	33, 38	eqtrdi 2242	. . . . . . . . . 10 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
40	39	eleq2d 2263	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( y e. ( { ( 1st ` y ) } X. w ) <-> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) )
41	39	sseq1d 3209	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( ( { ( 1st ` y ) } X. w ) C_ x <-> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x ) )
42	40, 41	anbi12d 473	. . . . . . . 8 |- ( w = { ( 2nd ` y ) } -> ( ( y e. ( { ( 1st ` y ) } X. w ) /\ ( { ( 1st ` y ) } X. w ) C_ x ) <-> ( y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } /\ { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x ) ) )
43	32, 42	rspc2ev 2880	. . . . . . 7 |- ( ( { ( 1st ` y ) } e. ~P A /\ { ( 2nd ` y ) } e. ~P B /\ ( y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } /\ { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x ) ) -> E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) )
44	17, 20, 25, 28, 43	syl112anc 1253	. . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) )
45	44	expr 375	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ y e. x ) -> ( x e. ~P ( A X. B ) -> E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
46	45	ralrimdva 2574	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( x e. ~P ( A X. B ) -> A. y e. x E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
47		eltx 14438	. . . . 5 |- ( ( ~P A e. Top /\ ~P B e. Top ) -> ( x e. ( ~P A tX ~P B ) <-> A. y e. x E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
48	1, 2, 47	syl2an 289	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( x e. ( ~P A tX ~P B ) <-> A. y e. x E. z e. ~P A E. w e. ~P B ( y e. ( z X. w ) /\ ( z X. w ) C_ x ) ) )
49	46, 48	sylibrd 169	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. ~P ( A X. B ) -> x e. ( ~P A tX ~P B ) ) )
50	49	ssrdv 3186	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) C_ ( ~P A tX ~P B ) )
51	12, 50	eqssd 3197	1 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   C_ wss 3154   ~Pcpw 3602   {csn 3619   <.cop 3622   U.cuni 3836   X. cxp 4658   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194   Topctop 14176   tX ctx 14431

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-topgen 12874  df-top 14177  df-topon 14190  df-bases 14222  df-tx 14432

This theorem is referenced by: (None)