Theorem unitg 21577

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 21574	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
2		velpw 4546	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵)
3	1, 2	sylibr 236	. . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵)
4	3	ssriv 3973	. . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵
5		sspwuni 5024	. . . 4 ⊢ ((topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵 ↔ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
6	4, 5	mpbi 232	. . 3 ⊢ ∪ (topGen‘𝐵) ⊆ ∪ 𝐵
7	6	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) ⊆ ∪ 𝐵)
8		bastg 21576	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
9	8	unissd 4850	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵))
10	7, 9	eqssd 3986	1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840  ‘cfv 6357  topGenctg 16713

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-topgen 16719

This theorem is referenced by:  tgcl  21579  tgtopon  21581  tgcmp  22011  2ndcsep  22069  txtopon  22201  ptuni  22204  xkouni  22209  prdstopn  22238  tgqtop  22322  alexsubb  22656  alexsubALTlem3  22659  alexsubALTlem4  22660  ptcmplem1  22662  uniretop  23373  fneval  33702  fnemeet1  33716  kelac2  39672