Theorem toprntopon 22828

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 22821	. . . . 5 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
2		fvex 6839	. . . . . 6 ⊢ (TopOn‘∪ 𝑥) ∈ V
3		eleq2 2817	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
4		eleq1 2816	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘∪ 𝑥) ∈ ran TopOn))
5	3, 4	anbi12d 632	. . . . . . 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 22827	. . . . . . . . . 10 ⊢ TopOn Fn V
8		vuniex 7679	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
9		fnfvelrn 7018	. . . . . . . . . 10 ⊢ ((TopOn Fn V ∧ ∪ 𝑥 ∈ V) → (TopOn‘∪ 𝑥) ∈ ran TopOn)
10	7, 8, 9	mp2an 692	. . . . . . . . 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 3563	. . . . 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 22823	. . . . . . . . 9 ⊢ Fun TopOn
17		elrnrexdm 7027	. . . . . . . . 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 3059	. . . . . . . 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 3996	. . . . . . . . . . 11 ⊢ (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
23	22	sseld 3936	. . . . . . . . . 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 1835	. . . . . . . 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 593	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
28		topontop 22816	. . . . . . 7 ⊢ (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
29	28	exlimiv 1930	. . . . . 6 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
30	27, 29	syl 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
31	30	exlimiv 1930	. . . 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 4864	. . 3 ⊢ (𝑥 ∈ ∪ ran TopOn ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
34	32, 33	bitr4i 278	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ ∪ ran TopOn)
35	34	eqriv 2726	1 ⊢ Top = ∪ ran TopOn

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3438  ∪ cuni 4861  dom cdm 5623  ran crn 5624  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  Topctop 22796  TopOnctopon 22813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-topon 22814

This theorem is referenced by: (None)