Theorem fiinopn 14886

Description: The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

Assertion

Ref	Expression
fiinopn	|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )

Proof of Theorem fiinopn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwg 3679	. . . . . . 7 |- ( A e. Fin -> ( A e. ~P J <-> A C_ J ) )
2		sseq1 3263	. . . . . . . . . . . . . 14 |- ( x = A -> ( x C_ J <-> A C_ J ) )
3		neeq1 2427	. . . . . . . . . . . . . 14 |- ( x = A -> ( x =/= (/) <-> A =/= (/) ) )
4		eleq1 2297	. . . . . . . . . . . . . 14 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
5	2, 3, 4	3anbi123d 1349	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) <-> ( A C_ J /\ A =/= (/) /\ A e. Fin ) ) )
6		inteq 3954	. . . . . . . . . . . . . . 15 |- ( x = A -> |^| x = |^| A )
7	6	eleq1d 2303	. . . . . . . . . . . . . 14 |- ( x = A -> ( |^| x e. J <-> |^| A e. J ) )
8	7	imbi2d 230	. . . . . . . . . . . . 13 |- ( x = A -> ( ( J e. Top -> |^| x e. J ) <-> ( J e. Top -> |^| A e. J ) ) )
9	5, 8	imbi12d 234	. . . . . . . . . . . 12 |- ( x = A -> ( ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> ( J e. Top -> |^| x e. J ) ) <-> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( J e. Top -> |^| A e. J ) ) ) )
10		sp 1560	. . . . . . . . . . . . . 14 |- ( A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) )
11	10	adantl 277	. . . . . . . . . . . . 13 |- ( ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) )
12		istopfin 14882	. . . . . . . . . . . . 13 |- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
13	11, 12	syl11 31	. . . . . . . . . . . 12 |- ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> ( J e. Top -> |^| x e. J ) )
14	9, 13	vtoclg 2877	. . . . . . . . . . 11 |- ( A e. ~P J -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( J e. Top -> |^| A e. J ) ) )
15	14	com12 30	. . . . . . . . . 10 |- ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) )
16	15	3exp 1229	. . . . . . . . 9 |- ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) ) ) )
17	16	com3r 79	. . . . . . . 8 |- ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) ) ) )
18	17	com4r 86	. . . . . . 7 |- ( A e. ~P J -> ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) )
19	1, 18	biimtrrdi 164	. . . . . 6 |- ( A e. Fin -> ( A C_ J -> ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) ) )
20	19	pm2.43a 51	. . . . 5 |- ( A e. Fin -> ( A C_ J -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) )
21	20	com4l 84	. . . 4 |- ( A C_ J -> ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( J e. Top -> |^| A e. J ) ) ) ) )
22	21	pm2.43i 49	. . 3 |- ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( J e. Top -> |^| A e. J ) ) ) )
23	22	3imp 1220	. 2 |- ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( J e. Top -> |^| A e. J ) )
24	23	com12 30	1 |- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2205   =/= wne 2414   C_ wss 3213   (/)c0 3510   ~Pcpw 3671   U.cuni 3916   |^|cint 3951   Fincfn 6977   Topctop 14879

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-er 6769  df-en 6978  df-fin 6980  df-top 14880

This theorem is referenced by: (None)