Theorem tgvalt 7615

Description: The topology generated by a basis.

Assertion

Ref	Expression
tgvalt	|- (B e. Bases -> (topGen` B) = {x | x (_ U.(B i^i P~x)})

Distinct variable group:   x,B

Proof of Theorem tgvalt

Step	Hyp	Ref	Expression
1		uniexg 2877	. . 3 |- (B e. Bases -> U.B e. V)
2		abssexg 2753	. . 3 |- (U.B e. V -> {x | (x (_ U.B /\ x (_ U.P~x)} e. V)
3		uniin 2524	. . . . . . 7 |- U.(B i^i P~x) (_ (U.B i^i U.P~x)
4		sstr 2075	. . . . . . 7 |- ((x (_ U.(B i^i P~x) /\ U.(B i^i P~x) (_ (U.B i^i U.P~x)) -> x (_ (U.B i^i U.P~x))
5	3, 4	mpan2 698	. . . . . 6 |- (x (_ U.(B i^i P~x) -> x (_ (U.B i^i U.P~x))
6		ssin 2235	. . . . . 6 |- ((x (_ U.B /\ x (_ U.P~x) <-> x (_ (U.B i^i U.P~x))
7	5, 6	sylibr 200	. . . . 5 |- (x (_ U.(B i^i P~x) -> (x (_ U.B /\ x (_ U.P~x))
8	7	ss2abi 2123	. . . 4 |- {x | x (_ U.(B i^i P~x)} (_ {x | (x (_ U.B /\ x (_ U.P~x)}
9		ssexg 2726	. . . 4 |- (({x | x (_ U.(B i^i P~x)} (_ {x | (x (_ U.B /\ x (_ U.P~x)} /\ {x | (x (_ U.B /\ x (_ U.P~x)} e. V) -> {x | x (_ U.(B i^i P~x)} e. V)
10	8, 9	mpan 697	. . 3 |- ({x | (x (_ U.B /\ x (_ U.P~x)} e. V -> {x | x (_ U.(B i^i P~x)} e. V)
11	1, 2, 10	3syl 20	. 2 |- (B e. Bases -> {x | x (_ U.(B i^i P~x)} e. V)
12		ineq1 2213	. . . . . 6 |- (y = B -> (y i^i P~x) = (B i^i P~x))
13	12	unieqd 2516	. . . . 5 |- (y = B -> U.(y i^i P~x) = U.(B i^i P~x))
14	13	sseq2d 2092	. . . 4 |- (y = B -> (x (_ U.(y i^i P~x) <-> x (_ U.(B i^i P~x)))
15	14	abbidv 1580	. . 3 |- (y = B -> {x | x (_ U.(y i^i P~x)} = {x | x (_ U.(B i^i P~x)})
16		df-topgen 7597	. . 3 |- topGen = {<.y, z>. | (y e. Bases /\ z = {x | x (_ U.(y i^i P~x)})}
17	15, 16	fvopab4g 3785	. 2 |- ((B e. Bases /\ {x | x (_ U.(B i^i P~x)} e. V) -> (topGen` B) = {x | x (_ U.(B i^i P~x)})
18	11, 17	mpdan 706	1 |- (B e. Bases -> (topGen` B) = {x | x (_ U.(B i^i P~x)})

Syntax hints:   -> wi 3   /\ wa 223   = 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:  tgval2t 7616  eltgt 7617

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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