Theorem eltgt 7617

Description: Membership in a topology generated by a basis.

Assertion

Ref	Expression
eltgt	|- (B e. Bases -> (A e. (topGen` B) <-> A (_ U.(B i^i P~A)))

Proof of Theorem eltgt

Step	Hyp	Ref	Expression
1		tgvalt 7615	. . 3 |- (B e. Bases -> (topGen` B) = {x | x (_ U.(B i^i P~x)})
2	1	eleq2d 1544	. 2 |- (B e. Bases -> (A e. (topGen` B) <-> A e. {x | x (_ U.(B i^i P~x)}))
3		elisset 1820	. . . 4 |- (A e. {x | x (_ U.(B i^i P~x)} -> A e. V)
4	3	adantl 390	. . 3 |- ((B e. Bases /\ A e. {x | x (_ U.(B i^i P~x)}) -> A e. V)
5		ssexg 2726	. . . . 5 |- ((A (_ U.(B i^i P~A) /\ U.(B i^i P~A) e. V) -> A e. V)
6		inex1g 2723	. . . . . 6 |- (B e. Bases -> (B i^i P~A) e. V)
7		uniexg 2877	. . . . . 6 |- ((B i^i P~A) e. V -> U.(B i^i P~A) e. V)
8	6, 7	syl 10	. . . . 5 |- (B e. Bases -> U.(B i^i P~A) e. V)
9	5, 8	sylan2 453	. . . 4 |- ((A (_ U.(B i^i P~A) /\ B e. Bases) -> A e. V)
10	9	ancoms 438	. . 3 |- ((B e. Bases /\ A (_ U.(B i^i P~A)) -> A e. V)
11		id 59	. . . . 5 |- (x = A -> x = A)
12		pweq 2407	. . . . . . 7 |- (x = A -> P~x = P~A)
13	12	ineq2d 2220	. . . . . 6 |- (x = A -> (B i^i P~x) = (B i^i P~A))
14	13	unieqd 2516	. . . . 5 |- (x = A -> U.(B i^i P~x) = U.(B i^i P~A))
15	11, 14	sseq12d 2093	. . . 4 |- (x = A -> (x (_ U.(B i^i P~x) <-> A (_ U.(B i^i P~A)))
16	15	elabg 1902	. . 3 |- (A e. V -> (A e. {x | x (_ U.(B i^i P~x)} <-> A (_ U.(B i^i P~A)))
17	4, 10, 16	pm5.21nd 682	. 2 |- (B e. Bases -> (A e. {x | x (_ U.(B i^i P~x)} <-> A (_ U.(B i^i P~A)))
18	2, 17	bitrd 530	1 |- (B e. Bases -> (A e. (topGen` B) <-> A (_ U.(B i^i P~A)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  {cab 1466  Vcvv 1814   i^i cin 2049   (_ wss 2050  P~cpw 2405  U.cuni 2507  ` cfv 3188  Basesctb 7592  topGenctg 7593

This theorem is referenced by:  bastgt 7621  unitgt 7622  eltopt 7628  tgsst 7635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-topgen 7597

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