Theorem neiuni 12801

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 12800	. . . 4 |- ( J e. Top -> ( S C_ X <-> X e. ( ( nei ` J ) ` S ) ) )
3	2	biimpa 294	. . 3 |- ( ( J e. Top /\ S C_ X ) -> X e. ( ( nei ` J ) ` S ) )
4		elssuni 3817	. . 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 12787	. . . . . 6 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> x C_ X )
7	6	ex 114	. . . . 5 |- ( J e. Top -> ( x e. ( ( nei ` J ) ` S ) -> x C_ X ) )
8	7	adantr 274	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( nei ` J ) ` S ) -> x C_ X ) )
9	8	ralrimiv 2538	. . 3 |- ( ( J e. Top /\ S C_ X ) -> A. x e. ( ( nei ` J ) ` S ) x C_ X )
10		unissb 3819	. . 3 |- ( U. ( ( nei ` J ) ` S ) C_ X <-> A. x e. ( ( nei ` J ) ` S ) x C_ X )
11	9, 10	sylibr 133	. 2 |- ( ( J e. Top /\ S C_ X ) -> U. ( ( nei ` J ) ` S ) C_ X )
12	5, 11	eqssd 3159	1 |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   U.cuni 3789   ` cfv 5188   Topctop 12635   neicnei 12778

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-nei 12779

This theorem is referenced by: (None)