Theorem toprntopon 23156

Description: A topology is the same thing as a topology on a set (variable-free version). (Contributed by BJ, 27-Apr-2021.)

Assertion

Ref	Expression
toprntopon	⊢ Top = ∪ ran TopOn

Proof of Theorem toprntopon

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

Step	Hyp	Ref	Expression
1		toptopon2 23149	. . . . 5 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
2		fvex 6895	. . . . . 6 ⊢ (TopOn‘∪ 𝑥) ∈ V
3		eleq2 2851	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
4		eleq1 2850	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘∪ 𝑥) ∈ ran TopOn))
5	3, 4	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn)))
6		simpl 488	. . . . . . . 8 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘∪ 𝑥))
7		fntopon 23155	. . . . . . . . . 10 ⊢ TopOn Fn V
8		vuniex 7745	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
9		fnfvelrn 7077	. . . . . . . . . 10 ⊢ ((TopOn Fn V ∧ ∪ 𝑥 ∈ V) → (TopOn‘∪ 𝑥) ∈ ran TopOn)
10	7, 8, 9	mp2an 705	. . . . . . . . 9 ⊢ (TopOn‘∪ 𝑥) ∈ ran TopOn
11	10	jctr 534	. . . . . . . 8 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn))
12	6, 11	impbii 212	. . . . . . 7 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
13	5, 12	bitrdi 290	. . . . . 6 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
14	2, 13	spcev 3563	. . . . 5 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
15	1, 14	sylbi 220	. . . 4 ⊢ (𝑥 ∈ Top → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
16		funtopon 23151	. . . . . . . . 9 ⊢ Fun TopOn
17		elrnrexdm 7086	. . . . . . . . 9 ⊢ (Fun TopOn → (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧)))
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧))
19		rexex 3094	. . . . . . . 8 ⊢ (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
20	18, 19	syl 18	. . . . . . 7 ⊢ (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
21		19.42v 1986	. . . . . . . 8 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
22		eqimss 3992	. . . . . . . . . . 11 ⊢ (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
23	22	sseld 3933	. . . . . . . . . 10 ⊢ (𝑦 = (TopOn‘𝑧) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (TopOn‘𝑧)))
24	23	impcom 413	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
25	24	eximi 1868	. . . . . . . 8 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
26	21, 25	sylbir 238	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
27	20, 26	sylan2 605	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
28		topontop 23144	. . . . . . 7 ⊢ (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
29	28	exlimiv 1963	. . . . . 6 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
30	27, 29	syl 18	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
31	30	exlimiv 1963	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
32	15, 31	impbii 212	. . 3 ⊢ (𝑥 ∈ Top ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
33		eluni 4873	. . 3 ⊢ (𝑥 ∈ ∪ ran TopOn ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
34	32, 33	bitr4i 281	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ ∪ ran TopOn)
35	34	eqriv 2759	1 ⊢ Top = ∪ ran TopOn

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453  ∪ cuni 4870  dom cdm 5659  ran crn 5660  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  Topctop 23124  TopOnctopon 23141

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-topon 23142

This theorem is used by: (None)