Theorem tgdom 12866

Description: A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)

Assertion

Ref	Expression
tgdom	⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Proof of Theorem tgdom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4166	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
2		inss1 3347	. . . . 5 ⊢ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵
3		vpwex 4165	. . . . . . 7 ⊢ 𝒫 𝑥 ∈ V
4	3	inex2 4124	. . . . . 6 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ V
5	4	elpw 3572	. . . . 5 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵 ↔ (𝐵 ∩ 𝒫 𝑥) ⊆ 𝐵)
6	2, 5	mpbir 145	. . . 4 ⊢ (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵
7	6	a1i 9	. . 3 ⊢ (𝑥 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝑥) ∈ 𝒫 𝐵)
8		unieq 3805	. . . . . . 7 ⊢ ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦))
9	8	adantl 275	. . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → ∪ (𝐵 ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑦))
10		eltg4i 12849	. . . . . . 7 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
11	10	ad2antrr 485	. . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
12		eltg4i 12849	. . . . . . 7 ⊢ (𝑦 ∈ (topGen‘𝐵) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦))
13	12	ad2antlr 486	. . . . . 6 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑦 = ∪ (𝐵 ∩ 𝒫 𝑦))
14	9, 11, 13	3eqtr4d 2213	. . . . 5 ⊢ (((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) ∧ (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦)) → 𝑥 = 𝑦)
15	14	ex 114	. . . 4 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) → 𝑥 = 𝑦))
16		pweq 3569	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦)
17	16	ineq2d 3328	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦))
18	15, 17	impbid1 141	. . 3 ⊢ ((𝑥 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ (topGen‘𝐵)) → ((𝐵 ∩ 𝒫 𝑥) = (𝐵 ∩ 𝒫 𝑦) ↔ 𝑥 = 𝑦))
19	7, 18	dom2 6753	. 2 ⊢ (𝒫 𝐵 ∈ V → (topGen‘𝐵) ≼ 𝒫 𝐵)
20	1, 19	syl 14	1 ⊢ (𝐵 ∈ 𝑉 → (topGen‘𝐵) ≼ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  Vcvv 2730   ∩ cin 3120   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796   class class class wbr 3989  ‘cfv 5198   ≼ cdom 6717  topGenctg 12594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-dom 6720  df-topgen 12600

This theorem is referenced by: (None)