Theorem unitg 22942

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 22939	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
2		velpw 4547	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐵 ↔ 𝑥 ⊆ ∪ 𝐵)
3	1, 2	sylibr 234	. . . . 5 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ 𝒫 ∪ 𝐵)
4	3	ssriv 3926	. . . 4 ⊢ (topGen‘𝐵) ⊆ 𝒫 ∪ 𝐵
5		sspwuni 5043	. . . 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 22941	. . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵))
9	8	unissd 4861	. 2 ⊢ (𝐵 ∈ 𝑉 → ∪ 𝐵 ⊆ ∪ (topGen‘𝐵))
10	7, 9	eqssd 3940	1 ⊢ (𝐵 ∈ 𝑉 → ∪ (topGen‘𝐵) = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  topGenctg 17391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-topgen 17397

This theorem is referenced by:  tgcl  22944  tgtopon  22946  tgcmp  23376  2ndcsep  23434  txtopon  23566  ptuni  23569  xkouni  23574  prdstopn  23603  tgqtop  23687  alexsubb  24021  alexsubALTlem3  24024  alexsubALTlem4  24025  ptcmplem1  24027  uniretop  24737  fneval  36550  fnemeet1  36564  kelac2  43511