Theorem fiinopn 12642

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 3567	. . . . . . 7 |- ( A e. Fin -> ( A e. ~P J <-> A C_ J ) )
2		sseq1 3165	. . . . . . . . . . . . . 14 |- ( x = A -> ( x C_ J <-> A C_ J ) )
3		neeq1 2349	. . . . . . . . . . . . . 14 |- ( x = A -> ( x =/= (/) <-> A =/= (/) ) )
4		eleq1 2229	. . . . . . . . . . . . . 14 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
5	2, 3, 4	3anbi123d 1302	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) <-> ( A C_ J /\ A =/= (/) /\ A e. Fin ) ) )
6		inteq 3827	. . . . . . . . . . . . . . 15 |- ( x = A -> |^| x = |^| A )
7	6	eleq1d 2235	. . . . . . . . . . . . . 14 |- ( x = A -> ( |^| x e. J <-> |^| A e. J ) )
8	7	imbi2d 229	. . . . . . . . . . . . 13 |- ( x = A -> ( ( J e. Top -> |^| x e. J ) <-> ( J e. Top -> |^| A e. J ) ) )
9	5, 8	imbi12d 233	. . . . . . . . . . . 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 1499	. . . . . . . . . . . . . 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 275	. . . . . . . . . . . . 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 12638	. . . . . . . . . . . . 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 2786	. . . . . . . . . . 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 1192	. . . . . . . . 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	syl6bir 163	. . . . . 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 1183	. 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 103   /\ w3a 968   A.wal 1341   = wceq 1343   e. wcel 2136   =/= wne 2336   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   U.cuni 3789   |^|cint 3824   Fincfn 6706   Topctop 12635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707  df-fin 6709  df-top 12636

This theorem is referenced by: (None)