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Theorem toprntopon 22297
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 22290 . . . . 5 (π‘₯ ∈ Top ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
2 fvex 6859 . . . . . 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 22296 . . . . . . . . . 10 TopOn Fn V
8 vuniex 7680 . . . . . . . . . 10 βˆͺ π‘₯ ∈ V
9 fnfvelrn 7035 . . . . . . . . . 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 3567 . . . . 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 22292 . . . . . . . . 9 Fun TopOn
17 elrnrexdm 7043 . . . . . . . . 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 3076 . . . . . . . 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 4004 . . . . . . . . . . 11 (𝑦 = (TopOnβ€˜π‘§) β†’ 𝑦 βŠ† (TopOnβ€˜π‘§))
2322sseld 3947 . . . . . . . . . 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 22285 . . . . . . 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 4872 . . 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 3070  Vcvv 3447  βˆͺ cuni 4869  dom cdm 5637  ran crn 5638  Fun wfun 6494   Fn wfn 6495  β€˜cfv 6500  Topctop 22265  TopOnctopon 22282
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-topon 22283
This theorem is referenced by: (None)
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