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Theorem toprntopon 20777
 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.
StepHypRef Expression
1 toptopon2 20771 . . . . . 6 (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘ 𝑥))
21biimpi 206 . . . . 5 (𝑥 ∈ Top → 𝑥 ∈ (TopOn‘ 𝑥))
3 fvex 6239 . . . . . 6 (TopOn‘ 𝑥) ∈ V
4 eleq2 2719 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑥𝑦𝑥 ∈ (TopOn‘ 𝑥)))
5 eleq1 2718 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘ 𝑥) ∈ ran TopOn))
64, 5anbi12d 747 . . . . . . 7 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn)))
7 simpl 472 . . . . . . . . 9 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘ 𝑥))
8 fntopon 20776 . . . . . . . . . . . 12 TopOn Fn V
9 vuniex 6996 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9pm3.2i 470 . . . . . . . . . . 11 (TopOn Fn V ∧ 𝑥 ∈ V)
11 fnfvelrn 6396 . . . . . . . . . . 11 ((TopOn Fn V ∧ 𝑥 ∈ V) → (TopOn‘ 𝑥) ∈ ran TopOn)
1210, 11ax-mp 5 . . . . . . . . . 10 (TopOn‘ 𝑥) ∈ ran TopOn
1312jctr 564 . . . . . . . . 9 (𝑥 ∈ (TopOn‘ 𝑥) → (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn))
147, 13impbii 199 . . . . . . . 8 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥))
1514a1i 11 . . . . . . 7 (𝑦 = (TopOn‘ 𝑥) → ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥)))
166, 15bitrd 268 . . . . . 6 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥)))
173, 16spcev 3331 . . . . 5 (𝑥 ∈ (TopOn‘ 𝑥) → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
182, 17syl 17 . . . 4 (𝑥 ∈ Top → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
19 funtopon 20773 . . . . . . . . . 10 Fun TopOn
20 elrnrexdm 6403 . . . . . . . . . 10 (Fun TopOn → (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧)))
2119, 20ax-mp 5 . . . . . . . . 9 (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧))
22 rexex 3031 . . . . . . . . 9 (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
2321, 22syl 17 . . . . . . . 8 (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
2423anim2i 592 . . . . . . 7 ((𝑥𝑦𝑦 ∈ ran TopOn) → (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
25 19.42v 1921 . . . . . . . . 9 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) ↔ (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
2625biimpri 218 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)))
27 eqimss 3690 . . . . . . . . . . . 12 (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
2827sseld 3635 . . . . . . . . . . 11 (𝑦 = (TopOn‘𝑧) → (𝑥𝑦𝑥 ∈ (TopOn‘𝑧)))
2928com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦 = (TopOn‘𝑧) → 𝑥 ∈ (TopOn‘𝑧)))
3029imp 444 . . . . . . . . 9 ((𝑥𝑦𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
3130eximi 1802 . . . . . . . 8 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
3226, 31syl 17 . . . . . . 7 ((𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
3324, 32syl 17 . . . . . 6 ((𝑥𝑦𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
34 topontop 20766 . . . . . . . 8 (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
3534eximi 1802 . . . . . . 7 (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → ∃𝑧 𝑥 ∈ Top)
36 ax5e 1881 . . . . . . 7 (∃𝑧 𝑥 ∈ Top → 𝑥 ∈ Top)
3735, 36syl 17 . . . . . 6 (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
3833, 37syl 17 . . . . 5 ((𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
3938exlimiv 1898 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
4018, 39impbii 199 . . 3 (𝑥 ∈ Top ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
41 eluni 4471 . . . 4 (𝑥 ran TopOn ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
4241bicomi 214 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) ↔ 𝑥 ran TopOn)
4340, 42bitri 264 . 2 (𝑥 ∈ Top ↔ 𝑥 ran TopOn)
4443eqriv 2648 1 Top = ran TopOn
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231  ∪ cuni 4468  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  Topctop 20746  TopOnctopon 20763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-topon 20764 This theorem is referenced by: (None)
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