Theorem fiinopn 14591

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 3634	. . . . . . 7 |- ( A e. Fin -> ( A e. ~P J <-> A C_ J ) )
2		sseq1 3224	. . . . . . . . . . . . . 14 |- ( x = A -> ( x C_ J <-> A C_ J ) )
3		neeq1 2391	. . . . . . . . . . . . . 14 |- ( x = A -> ( x =/= (/) <-> A =/= (/) ) )
4		eleq1 2270	. . . . . . . . . . . . . 14 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
5	2, 3, 4	3anbi123d 1325	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) <-> ( A C_ J /\ A =/= (/) /\ A e. Fin ) ) )
6		inteq 3902	. . . . . . . . . . . . . . 15 |- ( x = A -> |^| x = |^| A )
7	6	eleq1d 2276	. . . . . . . . . . . . . 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 1535	. . . . . . . . . . . . . 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 14587	. . . . . . . . . . . . 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 2838	. . . . . . . . . . 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 1205	. . . . . . . . 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 1196	. 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 981   A.wal 1371   = wceq 1373   e. wcel 2178   =/= wne 2378   C_ wss 3174   (/)c0 3468   ~Pcpw 3626   U.cuni 3864   |^|cint 3899   Fincfn 6850   Topctop 14584

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-er 6643  df-en 6851  df-fin 6853  df-top 14585

This theorem is referenced by: (None)