Theorem ptincpw 14912

Description: A product topology is included in the powerset of the cross product of the underlying sets. The cross product and the empty set are elements of a product topology.

Hypotheses

Ref	Expression
ptincpw.1	|- X = U.J
ptincpw.2	|- Y = U.K

Assertion

Ref	Expression
ptincpw	|- ((J e. Top /\ K e. Top) -> ({(/), (X X. Y)} C_ (J X.t K) /\ (J X.t K) C_ ~P(X X. Y)))

Proof of Theorem ptincpw

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- (J X.t K) = (J X.t K)
2		ptincpw.1	. . 3 |- X = U.J
3		ptincpw.2	. . 3 |- Y = U.K
4	1, 2, 3	txuni 8935	. 2 |- ((J e. Top /\ K e. Top) -> U.(J X.t K) = (X X. Y))
5		preq2 3099	. . . . . 6 |- ((X X. Y) = U.(J X.t K) -> {(/), (X X. Y)} = {(/), U.(J X.t K)})
6	5	eqcoms 1887	. . . . 5 |- (U.(J X.t K) = (X X. Y) -> {(/), (X X. Y)} = {(/), U.(J X.t K)})
7	6	sseq1d 2644	. . . 4 |- (U.(J X.t K) = (X X. Y) -> ({(/), (X X. Y)} C_ (J X.t K) <-> {(/), U.(J X.t K)} C_ (J X.t K)))
8		pweq 3036	. . . . . 6 |- ((X X. Y) = U.(J X.t K) -> ~P(X X. Y) = ~PU.(J X.t K))
9	8	sseq2d 2645	. . . . 5 |- ((X X. Y) = U.(J X.t K) -> ((J X.t K) C_ ~P(X X. Y) <-> (J X.t K) C_ ~PU.(J X.t K)))
10	9	eqcoms 1887	. . . 4 |- (U.(J X.t K) = (X X. Y) -> ((J X.t K) C_ ~P(X X. Y) <-> (J X.t K) C_ ~PU.(J X.t K)))
11	7, 10	anbi12d 690	. . 3 |- (U.(J X.t K) = (X X. Y) -> (({(/), (X X. Y)} C_ (J X.t K) /\ (J X.t K) C_ ~P(X X. Y)) <-> ({(/), U.(J X.t K)} C_ (J X.t K) /\ (J X.t K) C_ ~PU.(J X.t K))))
12	1	txtop 8934	. . . 4 |- ((J e. Top /\ K e. Top) -> (J X.t K) e. Top)
13		topindis 14859	. . . 4 |- ((J X.t K) e. Top -> ({(/), U.(J X.t K)} C_ (J X.t K) /\ (J X.t K) C_ ~PU.(J X.t K)))
14	12, 13	syl 12	. . 3 |- ((J e. Top /\ K e. Top) -> ({(/), U.(J X.t K)} C_ (J X.t K) /\ (J X.t K) C_ ~PU.(J X.t K)))
15	11, 14	syl5bir 227	. 2 |- (U.(J X.t K) = (X X. Y) -> ((J e. Top /\ K e. Top) -> ({(/), (X X. Y)} C_ (J X.t K) /\ (J X.t K) C_ ~P(X X. Y))))
16	4, 15	mpcom 60	1 |- ((J e. Top /\ K e. Top) -> ({(/), (X X. Y)} C_ (J X.t K) /\ (J X.t K) C_ ~P(X X. Y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {cpr 3045  U.cuni 3177   X. cxp 3984  (class class class)co 4884  Topctop 8857   X.t ctx 8930

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-topsp 8862  df-bases 8863  df-topgen 8864  df-tx 8931

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