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Theorem toprntopon 22427
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 = βˆͺ ran TopOn

Proof of Theorem toprntopon
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toptopon2 22420 . . . . 5 (π‘₯ ∈ Top ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
2 fvex 6905 . . . . . 6 (TopOnβ€˜βˆͺ π‘₯) ∈ V
3 eleq2 2823 . . . . . . . 8 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯)))
4 eleq1 2822 . . . . . . . 8 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ (𝑦 ∈ ran TopOn ↔ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn))
53, 4anbi12d 632 . . . . . . 7 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn)))
6 simpl 484 . . . . . . . 8 ((π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn) β†’ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
7 fntopon 22426 . . . . . . . . . 10 TopOn Fn V
8 vuniex 7729 . . . . . . . . . 10 βˆͺ π‘₯ ∈ V
9 fnfvelrn 7083 . . . . . . . . . 10 ((TopOn Fn V ∧ βˆͺ π‘₯ ∈ V) β†’ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn)
107, 8, 9mp2an 691 . . . . . . . . 9 (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn
1110jctr 526 . . . . . . . 8 (π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) β†’ (π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn))
126, 11impbii 208 . . . . . . 7 ((π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn) ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
135, 12bitrdi 287 . . . . . 6 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯)))
142, 13spcev 3597 . . . . 5 (π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) β†’ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
151, 14sylbi 216 . . . 4 (π‘₯ ∈ Top β†’ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
16 funtopon 22422 . . . . . . . . 9 Fun TopOn
17 elrnrexdm 7091 . . . . . . . . 9 (Fun TopOn β†’ (𝑦 ∈ ran TopOn β†’ βˆƒπ‘§ ∈ dom TopOn𝑦 = (TopOnβ€˜π‘§)))
1816, 17ax-mp 5 . . . . . . . 8 (𝑦 ∈ ran TopOn β†’ βˆƒπ‘§ ∈ dom TopOn𝑦 = (TopOnβ€˜π‘§))
19 rexex 3077 . . . . . . . 8 (βˆƒπ‘§ ∈ dom TopOn𝑦 = (TopOnβ€˜π‘§) β†’ βˆƒπ‘§ 𝑦 = (TopOnβ€˜π‘§))
2018, 19syl 17 . . . . . . 7 (𝑦 ∈ ran TopOn β†’ βˆƒπ‘§ 𝑦 = (TopOnβ€˜π‘§))
21 19.42v 1958 . . . . . . . 8 (βˆƒπ‘§(π‘₯ ∈ 𝑦 ∧ 𝑦 = (TopOnβ€˜π‘§)) ↔ (π‘₯ ∈ 𝑦 ∧ βˆƒπ‘§ 𝑦 = (TopOnβ€˜π‘§)))
22 eqimss 4041 . . . . . . . . . . 11 (𝑦 = (TopOnβ€˜π‘§) β†’ 𝑦 βŠ† (TopOnβ€˜π‘§))
2322sseld 3982 . . . . . . . . . 10 (𝑦 = (TopOnβ€˜π‘§) β†’ (π‘₯ ∈ 𝑦 β†’ π‘₯ ∈ (TopOnβ€˜π‘§)))
2423impcom 409 . . . . . . . . 9 ((π‘₯ ∈ 𝑦 ∧ 𝑦 = (TopOnβ€˜π‘§)) β†’ π‘₯ ∈ (TopOnβ€˜π‘§))
2524eximi 1838 . . . . . . . 8 (βˆƒπ‘§(π‘₯ ∈ 𝑦 ∧ 𝑦 = (TopOnβ€˜π‘§)) β†’ βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§))
2621, 25sylbir 234 . . . . . . 7 ((π‘₯ ∈ 𝑦 ∧ βˆƒπ‘§ 𝑦 = (TopOnβ€˜π‘§)) β†’ βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§))
2720, 26sylan2 594 . . . . . 6 ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§))
28 topontop 22415 . . . . . . 7 (π‘₯ ∈ (TopOnβ€˜π‘§) β†’ π‘₯ ∈ Top)
2928exlimiv 1934 . . . . . 6 (βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§) β†’ π‘₯ ∈ Top)
3027, 29syl 17 . . . . 5 ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ π‘₯ ∈ Top)
3130exlimiv 1934 . . . 4 (βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ π‘₯ ∈ Top)
3215, 31impbii 208 . . 3 (π‘₯ ∈ Top ↔ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
33 eluni 4912 . . 3 (π‘₯ ∈ βˆͺ ran TopOn ↔ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
3432, 33bitr4i 278 . 2 (π‘₯ ∈ Top ↔ π‘₯ ∈ βˆͺ ran TopOn)
3534eqriv 2730 1 Top = βˆͺ ran TopOn
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475  βˆͺ cuni 4909  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  β€˜cfv 6544  Topctop 22395  TopOnctopon 22412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-topon 22413
This theorem is referenced by: (None)
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