Theorem txdis 15003

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 14811	. . . . 5 |- ( A e. V -> ~P A e. Top )
2		distop 14811	. . . . 5 |- ( B e. W -> ~P B e. Top )
3		unipw 4309	. . . . . . 7 |- U. ~P A = A
4	3	eqcomi 2235	. . . . . 6 |- A = U. ~P A
5		unipw 4309	. . . . . . 7 |- U. ~P B = B
6	5	eqcomi 2235	. . . . . 6 |- B = U. ~P B
7	4, 6	txuni 14989	. . . . 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 3282	. . . 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 4055	. . 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 3664	. . . . . . . . 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 6328	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 1st ` y ) e. A )
16		snelpwi 4303	. . . . . . . 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 6329	. . . . . . . 8 |- ( y e. ( A X. B ) -> ( 2nd ` y ) e. B )
19		snelpwi 4303	. . . . . . . 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 3701	. . . . . . . 8 |- y e. { y }
22		1st2nd2 6338	. . . . . . . . . 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 3682	. . . . . . . 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 2320	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
26		simprl 531	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> y e. x )
27	23, 26	eqeltrrd 2309	. . . . . . . 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 3818	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( y e. x /\ x e. ~P ( A X. B ) ) ) -> { <. ( 1st ` y ) , ( 2nd ` y ) >. } C_ x )
29		xpeq1 4739	. . . . . . . . . 10 |- ( z = { ( 1st ` y ) } -> ( z X. w ) = ( { ( 1st ` y ) } X. w ) )
30	29	eleq2d 2301	. . . . . . . . 9 |- ( z = { ( 1st ` y ) } -> ( y e. ( z X. w ) <-> y e. ( { ( 1st ` y ) } X. w ) ) )
31	29	sseq1d 3256	. . . . . . . . 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 4740	. . . . . . . . . . 11 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) )
34		1stexg 6330	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 1st ` y ) e. _V )
35	34	elv 2806	. . . . . . . . . . . 12 |- ( 1st ` y ) e. _V
36		2ndexg 6331	. . . . . . . . . . . . 13 |- ( y e. _V -> ( 2nd ` y ) e. _V )
37	36	elv 2806	. . . . . . . . . . . 12 |- ( 2nd ` y ) e. _V
38	35, 37	xpsn 5824	. . . . . . . . . . 11 |- ( { ( 1st ` y ) } X. { ( 2nd ` y ) } ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. }
39	33, 38	eqtrdi 2280	. . . . . . . . . 10 |- ( w = { ( 2nd ` y ) } -> ( { ( 1st ` y ) } X. w ) = { <. ( 1st ` y ) , ( 2nd ` y ) >. } )
40	39	eleq2d 2301	. . . . . . . . 9 |- ( w = { ( 2nd ` y ) } -> ( y e. ( { ( 1st ` y ) } X. w ) <-> y e. { <. ( 1st ` y ) , ( 2nd ` y ) >. } ) )
41	39	sseq1d 3256	. . . . . . . . 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 2925	. . . . . . 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 1277	. . . . . 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 2612	. . . 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 14985	. . . . 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 3233	. 2 |- ( ( A e. V /\ B e. W ) -> ~P ( A X. B ) C_ ( ~P A tX ~P B ) )
51	12, 50	eqssd 3244	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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   {csn 3669   <.cop 3672   U.cuni 3893   X. cxp 4723   ` cfv 5326  (class class class)co 6018   1stc1st 6301   2ndc2nd 6302   Topctop 14723   tX ctx 14978

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-topgen 13344  df-top 14724  df-topon 14737  df-bases 14769  df-tx 14979

This theorem is referenced by: (None)