Theorem toprntopon 23063

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 23056	. . . . 5 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘∪ 𝑥))
2		fvex 6896	. . . . . 6 ⊢ (TopOn‘∪ 𝑥) ∈ V
3		eleq2 2852	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (TopOn‘∪ 𝑥)))
4		eleq1 2851	. . . . . . . 8 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘∪ 𝑥) ∈ ran TopOn))
5	3, 4	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = (TopOn‘∪ 𝑥) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn)))
6		simpl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ (TopOn‘∪ 𝑥) ∧ (TopOn‘∪ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘∪ 𝑥))
7		fntopon 23062	. . . . . . . . . 10 ⊢ TopOn Fn V
8		vuniex 7739	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ V
9		fnfvelrn 7077	. . . . . . . . . 10 ⊢ ((TopOn Fn V ∧ ∪ 𝑥 ∈ V) → (TopOn‘∪ 𝑥) ∈ ran TopOn)
10	7, 8, 9	mp2an 704	. . . . . . . . 9 ⊢ (TopOn‘∪ 𝑥) ∈ ran TopOn
11	10	jctr 533	. . . . . . . 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 3566	. . . . 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 23058	. . . . . . . . 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 3095	. . . . . . . 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 1983	. . . . . . . 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 3937	. . . . . . . . . 10 ⊢ (𝑦 = (TopOn‘𝑧) → (𝑥 ∈ 𝑦 → 𝑥 ∈ (TopOn‘𝑧)))
24	23	impcom 412	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
25	24	eximi 1865	. . . . . . . 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 604	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
28		topontop 23051	. . . . . . 7 ⊢ (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
29	28	exlimiv 1960	. . . . . 6 ⊢ (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
30	27, 29	syl 18	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
31	30	exlimiv 1960	. . . 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 4876	. . 3 ⊢ (𝑥 ∈ ∪ ran TopOn ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
34	32, 33	bitr4i 281	. 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ∈ ∪ ran TopOn)
35	34	eqriv 2760	1 ⊢ Top = ∪ ran TopOn

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  ∪ cuni 4873  dom cdm 5663  ran crn 5664  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  Topctop 23031  TopOnctopon 23048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-topon 23049

This theorem is referenced by: (None)