Theorem unitg 13647

Description: The topology generated by a basis 𝐵 is a topology on ∪ 𝐵. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)

Assertion

Ref	Expression
unitg	⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Proof of Theorem unitg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tg1 13644	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
2		velpw 3584	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵)
3	1, 2	sylibr 134	. . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵)
4	3	ssriv 3161	. . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵
5		sspwuni 3973	. . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 ↔ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
6	4, 5	mpbi 145	. . 3 ⊢ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵
7	6	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
8		bastg 13646	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
9	8	unissd 3835	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵))
10	7, 9	eqssd 3174	1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  𝒫 cpw 3577  ∪ cuni 3811  ‘cfv 5218  topGenctg 12708

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-topgen 12714

This theorem is referenced by:  tgcl  13649  tgtopon  13651  txtopon  13847  uniretop  14110