Theorem eltgt 7618

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 7616	. . 3 |- (B e. Bases -> (topGen` B) = {x | x (_ U.(B i^i P~x)})
2	1	eleq2d 1541	. 2 |- (B e. Bases -> (A e. (topGen` B) <-> A e. {x | x (_ U.(B i^i P~x)}))
3		elisset 1817	. . . 4 |- (A e. {x | x (_ U.(B i^i P~x)} -> A e. V)
4	3	adantl 388	. . 3 |- ((B e. Bases /\ A e. {x | x (_ U.(B i^i P~x)}) -> A e. V)
5		ssexg 2721	. . . . 5 |- ((A (_ U.(B i^i P~A) /\ U.(B i^i P~A) e. V) -> A e. V)
6		inex1g 2718	. . . . . 6 |- (B e. Bases -> (B i^i P~A) e. V)
7		uniexg 2871	. . . . . 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 451	. . . 4 |- ((A (_ U.(B i^i P~A) /\ B e. Bases) -> A e. V)
10	9	ancoms 436	. . 3 |- ((B e. Bases /\ A (_ U.(B i^i P~A)) -> A e. V)
11		id 59	. . . . 5 |- (x = A -> x = A)
12		pweq 2403	. . . . . . 7 |- (x = A -> P~x = P~A)
13	12	ineq2d 2217	. . . . . 6 |- (x = A -> (B i^i P~x) = (B i^i P~A))
14	13	unieqd 2512	. . . . 5 |- (x = A -> U.(B i^i P~x) = U.(B i^i P~A))
15	11, 14	sseq12d 2090	. . . 4 |- (x = A -> (x (_ U.(B i^i P~x) <-> A (_ U.(B i^i P~A)))
16	15	elabg 1899	. . 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 680	. 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 528	1 |- (B e. Bases -> (A e. (topGen` B) <-> A (_ U.(B i^i P~A)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811   i^i cin 2046   (_ wss 2047  P~cpw 2401  U.cuni 2503  ` cfv 3182  Basesctb 7590  topGenctg 7591

This theorem is referenced by:  bastgt 7622  unitgt 7623  eltopt 7629  tgsst 7636

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-topgen 7595

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