Theorem txdis 12917

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 12725	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		distop 12725	. . . . 5 |- ( B e. W -> ~P B e. Top )
3		unipw 4195	. . . . . . 7 |- U. ~P A = A
4	3	eqcomi 2169	. . . . . 6 |- A = U. ~P A
5		unipw 4195	. . . . . . 7 |- U. ~P B = B
6	5	eqcomi 2169	. . . . . 6 |- B = U. ~P B
7	4, 6	txuni 12903	. . . . 5 |- ( ( ~P A e. Top /\ ~P B e. Top ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
8	1, 2, 7	syl2an 287	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
9		eqimss2 3197	. . . 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 3950	. . 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 133	. 2 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) C_ ~P ( A X. B ) )
13		elelpwi 3571	. . . . . . . . 9 |- ( ( y e. x /\ x e. ~P ( A X. B ) ) -> y e. ( A X. B ) )
14	13	adantl 275	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. ( A X. B ) )
15		xp1st 6133	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 1st ` y ) e. A )
16		snelpwi 4190	. . . . . . . 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 6134	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 2nd ` y ) e. B )
19		snelpwi 4190	. . . . . . . 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 3608	. . . . . . . 8 |- y e. { y }
22		1st2nd2 6143	. . . . . . . . . 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 3589	. . . . . . . 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 2255	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
26		simprl 521	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. x )
27	23, 26	eqeltrrd 2244	. . . . . . . 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 3718	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x )
29		xpeq1 4618	. . . . . . . . . 10 |- ( z = { ( 1st ` y ) } -> ( z X. w ) = ( { ( 1st ` y ) } X. w ) )
30	29	eleq2d 2236	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( y e. ( z X. w ) <-> y e. ( { ( 1st ` y ) } X. w ) ) )
31	29	sseq1d 3171	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( ( z X. w ) C_ x <-> ( { ( 1st ` y ) } X. w ) C_ x ) )
32	30, 31	anbi12d 465	. . . . . . . 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 4619	. . . . . . . . . . 11 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) )
34		1stexg 6135	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
35	34	elv 2730	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
36		2ndexg 6136	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 2nd ` y ) e. _V )
37	36	elv 2730	. . . . . . . . . . . 12 |- ( 2nd ` y ) e. _V
38	35, 37	xpsn 5661	. . . . . . . . . . 11 |- ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. }
39	33, 38	eqtrdi 2215	. . . . . . . . . 10 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
40	39	eleq2d 2236	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( y e. ( { ( 1st ` y ) } X. w ) <-> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) )
41	39	sseq1d 3171	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( ( { ( 1st ` y ) } X. w ) C_ x <-> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x ) )
42	40, 41	anbi12d 465	. . . . . . . 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 2845	. . . . . . 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 1232	. . . . . 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 373	. . . . 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 2546	. . . 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 12899	. . . . 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 287	. . . 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 168	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. ~P ( A X. B ) -> x e. ( ~P A tX ~P B ) ) )
50	49	ssrdv 3148	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) C_ ( ~P A tX ~P B ) )
51	12, 50	eqssd 3159	1 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   {csn 3576   <.cop 3579   U.cuni 3789   X. cxp 4602   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Topctop 12635   tX ctx 12892

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-topgen 12577  df-top 12636  df-topon 12649  df-bases 12681  df-tx 12893

This theorem is referenced by: (None)