Theorem neiuni 14340

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 14339	. . . 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 3864	. . 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 14326	. . . . . 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 2566	. . 3 |- ( ( J e. Top /\ S C_ X ) -> A. x e. ( ( nei ` J ) ` S ) x C_ X )
10		unissb 3866	. . 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 3197	1 |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   C_ wss 3154   U.cuni 3836   ` cfv 5255   Topctop 14176   neicnei 14317

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 14177  df-nei 14318

This theorem is referenced by: (None)