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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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