Theorem neiuni 14884

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 14883	. . . 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 3921	. . 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 14870	. . . . . 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 2604	. . 3 |- ( ( J e. Top /\ S C_ X ) -> A. x e. ( ( nei ` J ) ` S ) x C_ X )
10		unissb 3923	. . 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 3244	1 |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   U.cuni 3893   ` cfv 5326   Topctop 14720   neicnei 14861

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-top 14721  df-nei 14862

This theorem is referenced by: (None)