Theorem neiuni 13232

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 13231	. . . 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 3833	. . 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 13218	. . . . . 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 2547	. . 3 |- ( ( J e. Top /\ S C_ X ) -> A. x e. ( ( nei ` J ) ` S ) x C_ X )
10		unissb 3835	. . 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 3170	1 |- ( ( J e. Top /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   A.wral 2453   C_ wss 3127   U.cuni 3805   ` cfv 5208   Topctop 13066   neicnei 13209

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-top 13067  df-nei 13210

This theorem is referenced by: (None)