Theorem neiuni 14829

Description: The union of the neighborhoods of a set equals the topology's underlying set. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 9-Apr-2015.)

Hypothesis

Ref	Expression
tpnei.1	|- X = U. J

Assertion

Ref	Expression
neiuni	|- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Proof of Theorem neiuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tpnei.1	. . . . 5 |- X = U. J
2	1	tpnei 14828	. . . 4 |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )
3	2	biimpa 296	. . 3 |- ( ( J e. Top /\ S C_ X ) -> X e. ( ( nei ` J ) ` S ) )
4		elssuni 3915	. . 3 |- ( X e. ( ( nei ` J ) ` S ) -> X C_ U. ( ( nei ` J ) ` S ) )
5	3, 4	syl 14	. 2 |- ( ( J e. Top /\ S C_ X ) -> X C_ U. ( ( nei ` J ) ` S ) )
6	1	neii1 14815	. . . . . 6 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> x C_ X )
7	6	ex 115	. . . . 5 |- ( J e. Top -> ( x e. ( ( nei ` J ) ` S ) -> x C_ X ) )
8	7	adantr 276	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( nei ` J ) ` S ) -> x C_ X ) )
9	8	ralrimiv 2602	. . 3 |- ( ( J e. Top /\ S C_ X ) -> A. x e. ( ( nei ` J ) ` S ) x C_ X )
10		unissb 3917	. . 3 |- ( U. ( ( nei ` J ) ` S ) C_ X <-> A. x e. ( ( nei ` J ) ` S ) x C_ X )
11	9, 10	sylibr 134	. 2 |- ( ( J e. Top /\ S C_ X ) -> U. ( ( nei ` J ) ` S ) C_ X )
12	5, 11	eqssd 3241	1 |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   U.cuni 3887   ` cfv 5317   Topctop 14665   neicnei 14806

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-top 14666  df-nei 14807

This theorem is referenced by: (None)