Theorem txdis 21435

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 20799	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		distop 20799	. . . . 5 |- ( B e. W -> ~P B e. Top )
3		unipw 4918	. . . . . . 7 |- U. ~P A = A
4	3	eqcomi 2631	. . . . . 6 |- A = U. ~P A
5		unipw 4918	. . . . . . 7 |- U. ~P B = B
6	5	eqcomi 2631	. . . . . 6 |- B = U. ~P B
7	4, 6	txuni 21395	. . . . 5 |- ( ( ~P A e. Top /\ ~P B e. Top ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
8	1, 2, 7	syl2an 494	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) = U. ( ~P A tX ~P B ) )
9		eqimss2 3658	. . . 4 |- ( ( A X. B ) = U. ( ~P A tX ~P B ) -> U. ( ~P A tX ~P B ) C_ ( A X. B ) )
10	8, 9	syl 17	. . 3 |- ( ( A e. V /\ B e. W ) -> U. ( ~P A tX ~P B ) C_ ( A X. B ) )
11		sspwuni 4611	. . 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 224	. 2 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) C_ ~P ( A X. B ) )
13		elelpwi 4171	. . . . . . . . 9 |- ( ( y e. x /\ x e. ~P ( A X. B ) ) -> y e. ( A X. B ) )
14	13	adantl 482	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. ( A X. B ) )
15		xp1st 7198	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 1st ` y ) e. A )
16		snelpwi 4912	. . . . . . . 8 |- ( ( 1st ` y ) e. A -> { ( 1st ` y ) } e. ~P A )
17	14, 15, 16	3syl 18	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { ( 1st ` y ) } e. ~P A )
18		xp2nd 7199	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 2nd ` y ) e. B )
19		snelpwi 4912	. . . . . . . 8 |- ( ( 2nd ` y ) e. B -> { ( 2nd ` y ) } e. ~P B )
20	14, 18, 19	3syl 18	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { ( 2nd ` y ) } e. ~P B )
21		vsnid 4209	. . . . . . . 8 |- y e. { y }
22		1st2nd2 7205	. . . . . . . . . 10 |- ( y e. ( A X. B ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
23	14, 22	syl 17	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
24	23	sneqd 4189	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { y } = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
25	21, 24	syl5eleq 2707	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
26		simprl 794	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. x )
27	23, 26	eqeltrrd 2702	. . . . . . . 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 4340	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x )
29		xpeq1 5128	. . . . . . . . . 10 |- ( z = { ( 1st ` y ) } -> ( z X. w ) = ( { ( 1st ` y ) } X. w ) )
30	29	eleq2d 2687	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( y e. ( z X. w ) <-> y e. ( { ( 1st ` y ) } X. w ) ) )
31	29	sseq1d 3632	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( ( z X. w ) C_ x <-> ( { ( 1st ` y ) } X. w ) C_ x ) )
32	30, 31	anbi12d 747	. . . . . . . 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 5129	. . . . . . . . . . 11 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) )
34		fvex 6201	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
35		fvex 6201	. . . . . . . . . . . 12 |- ( 2nd ` y ) e. _V
36	34, 35	xpsn 6407	. . . . . . . . . . 11 |- ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. }
37	33, 36	syl6eq 2672	. . . . . . . . . 10 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
38	37	eleq2d 2687	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( y e. ( { ( 1st ` y ) } X. w ) <-> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) )
39	37	sseq1d 3632	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( ( { ( 1st ` y ) } X. w ) C_ x <-> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x ) )
40	38, 39	anbi12d 747	. . . . . . . 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 ) ) )
41	32, 40	rspc2ev 3324	. . . . . . 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 ) )
42	17, 20, 25, 28, 41	syl112anc 1330	. . . . . 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 ) )
43	42	expr 643	. . . . 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 ) ) )
44	43	ralrimdva 2969	. . . 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 ) ) )
45		eltx 21371	. . . . 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 ) ) )
46	1, 2, 45	syl2an 494	. . . 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 ) ) )
47	44, 46	sylibrd 249	. . 3 |- ( ( A e. V /\ B e. W ) -> ( x e. ~P ( A X. B ) -> x e. ( ~P A tX ~P B ) ) )
48	47	ssrdv 3609	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) C_ ( ~P A tX ~P B ) )
49	12, 48	eqssd 3620	1 |- ( ( A e. V /\ B e. W ) -> ( ~P A tX ~P B ) = ~P ( A X. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Topctop 20698   tX ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-tx 21365

This theorem is referenced by:  distgp  21903