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Theorem toprntopon 22426
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 22419 . . . . 5 (π‘₯ ∈ Top ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
2 fvex 6904 . . . . . 6 (TopOnβ€˜βˆͺ π‘₯) ∈ V
3 eleq2 2822 . . . . . . . 8 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ (π‘₯ ∈ 𝑦 ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯)))
4 eleq1 2821 . . . . . . . 8 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ (𝑦 ∈ ran TopOn ↔ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn))
53, 4anbi12d 631 . . . . . . 7 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ (π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn)))
6 simpl 483 . . . . . . . 8 ((π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯) ∧ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn) β†’ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯))
7 fntopon 22425 . . . . . . . . . 10 TopOn Fn V
8 vuniex 7728 . . . . . . . . . 10 βˆͺ π‘₯ ∈ V
9 fnfvelrn 7082 . . . . . . . . . 10 ((TopOn Fn V ∧ βˆͺ π‘₯ ∈ V) β†’ (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn)
107, 8, 9mp2an 690 . . . . . . . . 9 (TopOnβ€˜βˆͺ π‘₯) ∈ ran TopOn
1110jctr 525 . . . . . . . 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 286 . . . . . 6 (𝑦 = (TopOnβ€˜βˆͺ π‘₯) β†’ ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) ↔ π‘₯ ∈ (TopOnβ€˜βˆͺ π‘₯)))
142, 13spcev 3596 . . . . 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 22421 . . . . . . . . 9 Fun TopOn
17 elrnrexdm 7090 . . . . . . . . 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 1957 . . . . . . . 8 (βˆƒπ‘§(π‘₯ ∈ 𝑦 ∧ 𝑦 = (TopOnβ€˜π‘§)) ↔ (π‘₯ ∈ 𝑦 ∧ βˆƒπ‘§ 𝑦 = (TopOnβ€˜π‘§)))
22 eqimss 4040 . . . . . . . . . . 11 (𝑦 = (TopOnβ€˜π‘§) β†’ 𝑦 βŠ† (TopOnβ€˜π‘§))
2322sseld 3981 . . . . . . . . . 10 (𝑦 = (TopOnβ€˜π‘§) β†’ (π‘₯ ∈ 𝑦 β†’ π‘₯ ∈ (TopOnβ€˜π‘§)))
2423impcom 408 . . . . . . . . 9 ((π‘₯ ∈ 𝑦 ∧ 𝑦 = (TopOnβ€˜π‘§)) β†’ π‘₯ ∈ (TopOnβ€˜π‘§))
2524eximi 1837 . . . . . . . 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 593 . . . . . 6 ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§))
28 topontop 22414 . . . . . . 7 (π‘₯ ∈ (TopOnβ€˜π‘§) β†’ π‘₯ ∈ Top)
2928exlimiv 1933 . . . . . 6 (βˆƒπ‘§ π‘₯ ∈ (TopOnβ€˜π‘§) β†’ π‘₯ ∈ Top)
3027, 29syl 17 . . . . 5 ((π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ π‘₯ ∈ Top)
3130exlimiv 1933 . . . 4 (βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn) β†’ π‘₯ ∈ Top)
3215, 31impbii 208 . . 3 (π‘₯ ∈ Top ↔ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
33 eluni 4911 . . 3 (π‘₯ ∈ βˆͺ ran TopOn ↔ βˆƒπ‘¦(π‘₯ ∈ 𝑦 ∧ 𝑦 ∈ ran TopOn))
3432, 33bitr4i 277 . 2 (π‘₯ ∈ Top ↔ π‘₯ ∈ βˆͺ ran TopOn)
3534eqriv 2729 1 Top = βˆͺ ran TopOn
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  βˆͺ cuni 4908  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  β€˜cfv 6543  Topctop 22394  TopOnctopon 22411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-topon 22412
This theorem is referenced by: (None)
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