Theorem unitg 22882

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 22879	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
2		velpw 4552	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵)
3	1, 2	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵)
4	3	ssriv 3933	. . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵
5		sspwuni 5046	. . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 ↔ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
6	4, 5	mpbi 230	. . 3 ⊢ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵
7	6	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
8		bastg 22881	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
9	8	unissd 4866	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵))
10	7, 9	eqssd 3947	1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4547  ∪ cuni 4856  ‘cfv 6481  topGenctg 17341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6437  df-fun 6483  df-fv 6489  df-topgen 17347

This theorem is referenced by:  tgcl  22884  tgtopon  22886  tgcmp  23316  2ndcsep  23374  txtopon  23506  ptuni  23509  xkouni  23514  prdstopn  23543  tgqtop  23627  alexsubb  23961  alexsubALTlem3  23964  alexsubALTlem4  23965  ptcmplem1  23967  uniretop  24677  fneval  36396  fnemeet1  36410  kelac2  43157