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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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