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Theorem toprntopon 20940
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 20933 . . . . . 6 (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘ 𝑥))
21biimpi 207 . . . . 5 (𝑥 ∈ Top → 𝑥 ∈ (TopOn‘ 𝑥))
3 fvex 6417 . . . . . 6 (TopOn‘ 𝑥) ∈ V
4 eleq2 2874 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑥𝑦𝑥 ∈ (TopOn‘ 𝑥)))
5 eleq1 2873 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘ 𝑥) ∈ ran TopOn))
64, 5anbi12d 618 . . . . . . 7 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn)))
7 simpl 470 . . . . . . . . 9 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘ 𝑥))
8 fntopon 20939 . . . . . . . . . . . 12 TopOn Fn V
9 vuniex 7180 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9pm3.2i 458 . . . . . . . . . . 11 (TopOn Fn V ∧ 𝑥 ∈ V)
11 fnfvelrn 6574 . . . . . . . . . . 11 ((TopOn Fn V ∧ 𝑥 ∈ V) → (TopOn‘ 𝑥) ∈ ran TopOn)
1210, 11ax-mp 5 . . . . . . . . . 10 (TopOn‘ 𝑥) ∈ ran TopOn
1312jctr 516 . . . . . . . . 9 (𝑥 ∈ (TopOn‘ 𝑥) → (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn))
147, 13impbii 200 . . . . . . . 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 270 . . . . . 6 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥)))
173, 16spcev 3493 . . . . 5 (𝑥 ∈ (TopOn‘ 𝑥) → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
182, 17syl 17 . . . 4 (𝑥 ∈ Top → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
19 funtopon 20935 . . . . . . . . . 10 Fun TopOn
20 elrnrexdm 6581 . . . . . . . . . 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 3189 . . . . . . . . 9 (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
2321, 22syl 17 . . . . . . . 8 (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
2423anim2i 605 . . . . . . 7 ((𝑥𝑦𝑦 ∈ ran TopOn) → (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
25 19.42v 2044 . . . . . . . . 9 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) ↔ (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
2625biimpri 219 . . . . . . . 8 ((𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)))
27 eqimss 3854 . . . . . . . . . . . 12 (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
2827sseld 3797 . . . . . . . . . . 11 (𝑦 = (TopOn‘𝑧) → (𝑥𝑦𝑥 ∈ (TopOn‘𝑧)))
2928com12 32 . . . . . . . . . 10 (𝑥𝑦 → (𝑦 = (TopOn‘𝑧) → 𝑥 ∈ (TopOn‘𝑧)))
3029imp 395 . . . . . . . . 9 ((𝑥𝑦𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
3130eximi 1919 . . . . . . . 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 20928 . . . . . . . 8 (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
3534eximi 1919 . . . . . . 7 (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → ∃𝑧 𝑥 ∈ Top)
36 ax5e 2003 . . . . . . 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 2021 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
4018, 39impbii 200 . . 3 (𝑥 ∈ Top ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
41 eluni 4633 . . . 4 (𝑥 ran TopOn ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
4241bicomi 215 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) ↔ 𝑥 ran TopOn)
4340, 42bitri 266 . 2 (𝑥 ∈ Top ↔ 𝑥 ran TopOn)
4443eqriv 2803 1 Top = ran TopOn
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2156  wrex 3097  Vcvv 3391   cuni 4630  dom cdm 5311  ran crn 5312  Fun wfun 6091   Fn wfn 6092  cfv 6097  Topctop 20908  TopOnctopon 20925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105  df-topon 20926
This theorem is referenced by: (None)
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