Theorem unitg 22883

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 22880	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
2		velpw 4554	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵)
3	1, 2	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵)
4	3	ssriv 3934	. . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵
5		sspwuni 5050	. . . 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 22882	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
9	8	unissd 4868	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵))
10	7, 9	eqssd 3948	1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ⊆ wss 3898  𝒫 cpw 4549  ∪ cuni 4858  ‘cfv 6486  topGenctg 17343

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-topgen 17349

This theorem is referenced by:  tgcl  22885  tgtopon  22887  tgcmp  23317  2ndcsep  23375  txtopon  23507  ptuni  23510  xkouni  23515  prdstopn  23544  tgqtop  23628  alexsubb  23962  alexsubALTlem3  23965  alexsubALTlem4  23966  ptcmplem1  23968  uniretop  24678  fneval  36417  fnemeet1  36431  kelac2  43182