Theorem fiinopn 12796

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 3574	. . . . . . 7 |- ( A e. Fin -> ( A e. ~P J <-> A C_ J ) )
2		sseq1 3170	. . . . . . . . . . . . . 14 |- ( x = A -> ( x C_ J <-> A C_ J ) )
3		neeq1 2353	. . . . . . . . . . . . . 14 |- ( x = A -> ( x =/= (/) <-> A =/= (/) ) )
4		eleq1 2233	. . . . . . . . . . . . . 14 |- ( x = A -> ( x e. Fin <-> A e. Fin ) )
5	2, 3, 4	3anbi123d 1307	. . . . . . . . . . . . 13 |- ( x = A -> ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) <-> ( A C_ J /\ A =/= (/) /\ A e. Fin ) ) )
6		inteq 3834	. . . . . . . . . . . . . . 15 |- ( x = A -> |^| x = |^| A )
7	6	eleq1d 2239	. . . . . . . . . . . . . 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 1504	. . . . . . . . . . . . . 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 12792	. . . . . . . . . . . . 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 2790	. . . . . . . . . . 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 1197	. . . . . . . . 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 1188	. 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 973   A.wal 1346   = wceq 1348   e. wcel 2141   =/= wne 2340   C_ wss 3121   (/)c0 3414   ~Pcpw 3566   U.cuni 3796   |^|cint 3831   Fincfn 6718   Topctop 12789

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721  df-top 12790

This theorem is referenced by: (None)