Theorem fiinopn 20706

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 4166	. . . . . . 7 |- ( A e. Fin -> ( A e. ~P J <-> A C_ J ) )
2		sseq1 3626	. . . . . . . . . . . . . 14 |- ( x = A -> ( x C_ J <-> A C_ J ) )
3		neeq1 2856	. . . . . . . . . . . . . 14 |- ( x = A -> ( x =/= (/) <-> A =/= (/) ) )
4		eleq1 2689	. . . . . . . . . . . . . 14 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
5	2, 3, 4	3anbi123d 1399	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) <-> ( A C_ J /\ A =/= (/) /\ A e. Fin ) ) )
6		inteq 4478	. . . . . . . . . . . . . . 15 |- ( x = A -> |^| x = |^| A )
7	6	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( x = A -> ( |^| x e. J <-> |^| A e. J ) )
8	7	imbi2d 330	. . . . . . . . . . . . 13 |- ( x = A -> ( ( J e. Top -> |^| x e. J ) <-> ( J e. Top -> |^| A e. J ) ) )
9	5, 8	imbi12d 334	. . . . . . . . . . . 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 2053	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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		istop2g 20701	. . . . . . . . . . . . . 14 |- ( J e. Top -> ( 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	12	ibi 256	. . . . . . . . . . . . 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 ) ) )
14	11, 13	syl11 33	. . . . . . . . . . . 12 |- ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> ( J e. Top -> |^| x e. J ) )
15	9, 14	vtoclg 3266	. . . . . . . . . . 11 |- ( A e. ~P J -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( J e. Top -> |^| A e. J ) ) )
16	15	com12 32	. . . . . . . . . 10 |- ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) )
17	16	3exp 1264	. . . . . . . . 9 |- ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) ) ) )
18	17	com3r 87	. . . . . . . 8 |- ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( A e. ~P J -> ( J e. Top -> |^| A e. J ) ) ) ) )
19	18	com4r 94	. . . . . . 7 |- ( A e. ~P J -> ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) )
20	1, 19	syl6bir 244	. . . . . 6 |- ( A e. Fin -> ( A C_ J -> ( A e. Fin -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) ) )
21	20	pm2.43a 54	. . . . 5 |- ( A e. Fin -> ( A C_ J -> ( A C_ J -> ( A =/= (/) -> ( J e. Top -> |^| A e. J ) ) ) ) )
22	21	com4l 92	. . . 4 |- ( A C_ J -> ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( J e. Top -> |^| A e. J ) ) ) ) )
23	22	pm2.43i 52	. . 3 |- ( A C_ J -> ( A =/= (/) -> ( A e. Fin -> ( J e. Top -> |^| A e. J ) ) ) )
24	23	3imp 1256	. 2 |- ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> ( J e. Top -> |^| A e. J ) )
25	24	com12 32	1 |- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   Fincfn 7955   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-top 20699

This theorem is referenced by:  iinopn  20707  hauscmplem  21209  1stcfb  21248  txtube  21443