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Theorem bj-toprntopon 32697
 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.
StepHypRef Expression
1 bj-toptopon2 32687 . . . . . 6 (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘ 𝑥))
21biimpi 206 . . . . 5 (𝑥 ∈ Top → 𝑥 ∈ (TopOn‘ 𝑥))
3 fvex 6158 . . . . . 6 (TopOn‘ 𝑥) ∈ V
4 eleq2 2687 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑥𝑦𝑥 ∈ (TopOn‘ 𝑥)))
5 eleq1 2686 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘ 𝑥) ∈ ran TopOn))
64, 5anbi12d 746 . . . . . . 7 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn)))
7 simpl 473 . . . . . . . . 9 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘ 𝑥))
8 bj-fntopon 32696 . . . . . . . . . . . 12 TopOn Fn V
9 vuniex 6907 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9pm3.2i 471 . . . . . . . . . . 11 (TopOn Fn V ∧ 𝑥 ∈ V)
11 fnfvelrn 6312 . . . . . . . . . . 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 3286 . . . . 5 (𝑥 ∈ (TopOn‘ 𝑥) → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
182, 17syl 17 . . . 4 (𝑥 ∈ Top → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
19 bj-funtopon 32689 . . . . . . . . . 10 Fun TopOn
20 elrnrexdm 6319 . . . . . . . . . 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 2996 . . . . . . . . 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 1915 . . . . . . . . 9 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) ↔ (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
2625biimpri 218 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)))
27 eqimss 3636 . . . . . . . . . . . 12 (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
2827sseld 3582 . . . . . . . . . . 11 (𝑦 = (TopOn‘𝑧) → (𝑥𝑦𝑥 ∈ (TopOn‘𝑧)))
2928com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦 = (TopOn‘𝑧) → 𝑥 ∈ (TopOn‘𝑧)))
3029imp 445 . . . . . . . . 9 ((𝑥𝑦𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
3130eximi 1759 . . . . . . . 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 20641 . . . . . . . 8 (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
3534eximi 1759 . . . . . . 7 (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → ∃𝑧 𝑥 ∈ Top)
36 ax5e 1838 . . . . . . 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 1855 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
4018, 39impbii 199 . . 3 (𝑥 ∈ Top ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
41 eluni 4405 . . . 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 2618 1 Top = ran TopOn
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃wrex 2908  Vcvv 3186  ∪ cuni 4402  dom cdm 5074  ran crn 5075  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847  Topctop 20617  TopOnctopon 20618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-topon 20623 This theorem is referenced by: (None)
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