Theorem toprntopon 22952

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 22945	. . . . 5 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
2		fvex 6933	. . . . . 6 ⊢ (TopOn‘∪ 𝑥) ∈ V
3		eleq2 2833	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
4		eleq1 2832	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘∪ 𝑥) ∈ ran TopOn))
5	3, 4	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn)))
6		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘∪ 𝑥))
7		fntopon 22951	. . . . . . . . . 10 ⊢ TopOn Fn V
8		vuniex 7774	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
9		fnfvelrn 7114	. . . . . . . . . 10 ⊢ ((TopOn Fn V ∧ ∪ 𝑥 ∈ V) → (TopOn‘∪ 𝑥) ∈ ran TopOn)
10	7, 8, 9	mp2an 691	. . . . . . . . 9 ⊢ (TopOn‘∪ 𝑥) ∈ ran TopOn
11	10	jctr 524	. . . . . . . 8 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn))
12	6, 11	impbii 209	. . . . . . 7 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
13	5, 12	bitrdi 287	. . . . . 6 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
14	2, 13	spcev 3619	. . . . 5 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
15	1, 14	sylbi 217	. . . 4 ⊢ (𝑥 ∈ Top → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
16		funtopon 22947	. . . . . . . . 9 ⊢ Fun TopOn
17		elrnrexdm 7123	. . . . . . . . 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 3082	. . . . . . . 8 ⊢ (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
21		19.42v 1953	. . . . . . . 8 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
22		eqimss 4067	. . . . . . . . . . 11 ⊢ (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
23	22	sseld 4007	. . . . . . . . . 10 ⊢ (𝑦 = (TopOn‘𝑧) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (TopOn‘𝑧)))
24	23	impcom 407	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
25	24	eximi 1833	. . . . . . . 8 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
26	21, 25	sylbir 235	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
27	20, 26	sylan2 592	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
28		topontop 22940	. . . . . . 7 ⊢ (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
29	28	exlimiv 1929	. . . . . 6 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
30	27, 29	syl 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
31	30	exlimiv 1929	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
32	15, 31	impbii 209	. . 3 ⊢ (𝑥 ∈ Top ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
33		eluni 4934	. . 3 ⊢ (𝑥 ∈ ∪ ran TopOn ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
34	32, 33	bitr4i 278	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ ∪ ran TopOn)
35	34	eqriv 2737	1 ⊢ Top = ∪ ran TopOn

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488  ∪ cuni 4931  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-topon 22938

This theorem is referenced by: (None)