MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  toprntopon Structured version   Visualization version   GIF version

Theorem toprntopon 22947
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 22940 . . . . 5 (𝑥 ∈ Top ↔ 𝑥 ∈ (TopOn‘ 𝑥))
2 fvex 6920 . . . . . 6 (TopOn‘ 𝑥) ∈ V
3 eleq2 2828 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑥𝑦𝑥 ∈ (TopOn‘ 𝑥)))
4 eleq1 2827 . . . . . . . 8 (𝑦 = (TopOn‘ 𝑥) → (𝑦 ∈ ran TopOn ↔ (TopOn‘ 𝑥) ∈ ran TopOn))
53, 4anbi12d 632 . . . . . . 7 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn)))
6 simpl 482 . . . . . . . 8 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) → 𝑥 ∈ (TopOn‘ 𝑥))
7 fntopon 22946 . . . . . . . . . 10 TopOn Fn V
8 vuniex 7758 . . . . . . . . . 10 𝑥 ∈ V
9 fnfvelrn 7100 . . . . . . . . . 10 ((TopOn Fn V ∧ 𝑥 ∈ V) → (TopOn‘ 𝑥) ∈ ran TopOn)
107, 8, 9mp2an 692 . . . . . . . . 9 (TopOn‘ 𝑥) ∈ ran TopOn
1110jctr 524 . . . . . . . 8 (𝑥 ∈ (TopOn‘ 𝑥) → (𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn))
126, 11impbii 209 . . . . . . 7 ((𝑥 ∈ (TopOn‘ 𝑥) ∧ (TopOn‘ 𝑥) ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥))
135, 12bitrdi 287 . . . . . 6 (𝑦 = (TopOn‘ 𝑥) → ((𝑥𝑦𝑦 ∈ ran TopOn) ↔ 𝑥 ∈ (TopOn‘ 𝑥)))
142, 13spcev 3606 . . . . 5 (𝑥 ∈ (TopOn‘ 𝑥) → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
151, 14sylbi 217 . . . 4 (𝑥 ∈ Top → ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
16 funtopon 22942 . . . . . . . . 9 Fun TopOn
17 elrnrexdm 7109 . . . . . . . . 9 (Fun TopOn → (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧)))
1816, 17ax-mp 5 . . . . . . . 8 (𝑦 ∈ ran TopOn → ∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧))
19 rexex 3074 . . . . . . . 8 (∃𝑧 ∈ dom TopOn𝑦 = (TopOn‘𝑧) → ∃𝑧 𝑦 = (TopOn‘𝑧))
2018, 19syl 17 . . . . . . 7 (𝑦 ∈ ran TopOn → ∃𝑧 𝑦 = (TopOn‘𝑧))
21 19.42v 1951 . . . . . . . 8 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) ↔ (𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)))
22 eqimss 4054 . . . . . . . . . . 11 (𝑦 = (TopOn‘𝑧) → 𝑦 ⊆ (TopOn‘𝑧))
2322sseld 3994 . . . . . . . . . 10 (𝑦 = (TopOn‘𝑧) → (𝑥𝑦𝑥 ∈ (TopOn‘𝑧)))
2423impcom 407 . . . . . . . . 9 ((𝑥𝑦𝑦 = (TopOn‘𝑧)) → 𝑥 ∈ (TopOn‘𝑧))
2524eximi 1832 . . . . . . . 8 (∃𝑧(𝑥𝑦𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
2621, 25sylbir 235 . . . . . . 7 ((𝑥𝑦 ∧ ∃𝑧 𝑦 = (TopOn‘𝑧)) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
2720, 26sylan2 593 . . . . . 6 ((𝑥𝑦𝑦 ∈ ran TopOn) → ∃𝑧 𝑥 ∈ (TopOn‘𝑧))
28 topontop 22935 . . . . . . 7 (𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
2928exlimiv 1928 . . . . . 6 (∃𝑧 𝑥 ∈ (TopOn‘𝑧) → 𝑥 ∈ Top)
3027, 29syl 17 . . . . 5 ((𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
3130exlimiv 1928 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn) → 𝑥 ∈ Top)
3215, 31impbii 209 . . 3 (𝑥 ∈ Top ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
33 eluni 4915 . . 3 (𝑥 ran TopOn ↔ ∃𝑦(𝑥𝑦𝑦 ∈ ran TopOn))
3432, 33bitr4i 278 . 2 (𝑥 ∈ Top ↔ 𝑥 ran TopOn)
3534eqriv 2732 1 Top = ran TopOn
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478   cuni 4912  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  cfv 6563  Topctop 22915  TopOnctopon 22932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-topon 22933
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator