Theorem bj-toprntopon 32244

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
bj-toprntopon	⊢ Top = ∪ ran TopOn

Proof of Theorem bj-toprntopon

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

Step	Hyp	Ref	Expression
1		bj-toptopon2 32234	. . . . . 6 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
2	1	biimpi 205	. . . . 5 ⊢ (𝑥 ∈ Top → 𝑥 ∈ (TopOn‘∪ 𝑥))
3		fvex 6113	. . . . . 6 ⊢ (TopOn‘∪ 𝑥) ∈ V
4		eleq2 2677	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
5		eleq1 2676	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘∪ 𝑥) ∈ ran TopOn))
6	4, 5	anbi12d 743	. . . . . . 7 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn)))
7		simpl 472	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘∪ 𝑥))
8		bj-fntopon 32243	. . . . . . . . . . . 12 ⊢ TopOn Fn V
9		vuniex 6852	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ V
10	8, 9	pm3.2i 470	. . . . . . . . . . 11 ⊢ (TopOn Fn V ∧ ∪ 𝑥 ∈ V)
11		fnfvelrn 6264	. . . . . . . . . . 11 ⊢ ((TopOn Fn V ∧ ∪ 𝑥 ∈ V) → (TopOn‘∪ 𝑥) ∈ ran TopOn)
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ (TopOn‘∪ 𝑥) ∈ ran TopOn
13	12	jctr 563	. . . . . . . . 9 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn))
14	7, 13	impbii 198	. . . . . . . 8 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
15	14	a1i 11	. . . . . . 7 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
16	6, 15	bitrd 267	. . . . . 6 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
17	3, 16	spcev 3273	. . . . 5 ⊢ (𝑥 ∈ (TopOn‘∪ 𝑥) → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
18	2, 17	syl 17	. . . 4 ⊢ (𝑥 ∈ Top → ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
19		bj-funtopon 32236	. . . . . . . . . 10 ⊢ Fun TopOn
20		elrnrexdm 6271	. . . . . . . . . 10 ⊢ (Fun TopOn → (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧)))
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧))
22		rexex 2985	. . . . . . . . 9 ⊢ (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
24	23	anim2i 591	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → (𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
25		19.42v 1905	. . . . . . . . 9 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
26	25	biimpri 217	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)))
27		eqimss 3620	. . . . . . . . . . . 12 ⊢ (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
28	27	sseld 3567	. . . . . . . . . . 11 ⊢ (𝑦 = (TopOn‘𝑧) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (TopOn‘𝑧)))
29	28	com12 32	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → (𝑦 = (TopOn‘𝑧) → 𝑥 ∈ (TopOn‘𝑧)))
30	29	imp 444	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
31	30	eximi 1752	. . . . . . . 8 ⊢ (∃𝑧(𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
32	26, 31	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
33	24, 32	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
34		topontop 20541	. . . . . . . 8 ⊢ (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
35	34	eximi 1752	. . . . . . 7 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → ∃𝑧 𝑥 ∈ Top)
36		ax5e 1829	. . . . . . 7 ⊢ (∃𝑧 𝑥 ∈ Top → 𝑥 ∈ Top)
37	35, 36	syl 17	. . . . . 6 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
38	33, 37	syl 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
39	38	exlimiv 1845	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
40	18, 39	impbii 198	. . 3 ⊢ (𝑥 ∈ Top ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
41		eluni 4375	. . . 4 ⊢ (𝑥 ∈ ∪ ran TopOn ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
42	41	bicomi 213	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ ∪ ran TopOn)
43	40, 42	bitri 263	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ ∪ ran TopOn)
44	43	eqriv 2607	1 ⊢ Top = ∪ ran TopOn

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ∪ cuni 4372  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Topctop 20517  TopOnctopon 20518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-topon 20523

This theorem is referenced by: (None)