Theorem txdis 13357

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 13165	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		distop 13165	. . . . 5 |- ( B e. W -> ~P B e. Top )
3		unipw 4211	. . . . . . 7 |- U. ~P A = A
4	3	eqcomi 2179	. . . . . 6 |- A = U. ~P A
5		unipw 4211	. . . . . . 7 |- U. ~P B = B
6	5	eqcomi 2179	. . . . . 6 |- B = U. ~P B
7	4, 6	txuni 13343	. . . . 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 3208	. . . 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 3966	. . 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 3584	. . . . . . . . 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 6156	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 1st ` y ) e. A )
16		snelpwi 4206	. . . . . . . 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 6157	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 2nd ` y ) e. B )
19		snelpwi 4206	. . . . . . . 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 3621	. . . . . . . 8 |- y e. { y }
22		1st2nd2 6166	. . . . . . . . . 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 3602	. . . . . . . 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 2264	. . . . . . 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 2253	. . . . . . . 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 3734	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x )
29		xpeq1 4634	. . . . . . . . . 10 |- ( z = { ( 1st ` y ) } -> ( z X. w ) = ( { ( 1st ` y ) } X. w ) )
30	29	eleq2d 2245	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( y e. ( z X. w ) <-> y e. ( { ( 1st ` y ) } X. w ) ) )
31	29	sseq1d 3182	. . . . . . . . 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 4635	. . . . . . . . . . 11 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) )
34		1stexg 6158	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
35	34	elv 2739	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
36		2ndexg 6159	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 2nd ` y ) e. _V )
37	36	elv 2739	. . . . . . . . . . . 12 |- ( 2nd ` y ) e. _V
38	35, 37	xpsn 5684	. . . . . . . . . . 11 |- ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. }
39	33, 38	eqtrdi 2224	. . . . . . . . . 10 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
40	39	eleq2d 2245	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( y e. ( { ( 1st ` y ) } X. w ) <-> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) )
41	39	sseq1d 3182	. . . . . . . . 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 2854	. . . . . . 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 1242	. . . . . 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 2555	. . . 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 13339	. . . . 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 3159	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) C_ ( ~P A tX ~P B ) )
51	12, 50	eqssd 3170	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 1353   e. wcel 2146   A.wral 2453   E.wrex 2454   _Vcvv 2735   C_ wss 3127   ~Pcpw 3572   {csn 3589   <.cop 3592   U.cuni 3805   X. cxp 4618   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Topctop 13075   tX ctx 13332

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-topgen 12640  df-top 13076  df-topon 13089  df-bases 13121  df-tx 13333

This theorem is referenced by: (None)