Theorem toprntopon 20729

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 = U. ran TopOn

Proof of Theorem toprntopon

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toptopon2 20723	. . . . . 6 |- ( x e. Top <-> x e. (TopOn ` U. x ) )
2	1	biimpi 206	. . . . 5 |- ( x e. Top -> x e. (TopOn ` U. x ) )
3		fvex 6201	. . . . . 6 |- (TopOn ` U. x ) e. _V
4		eleq2 2690	. . . . . . . 8 |- ( y = (TopOn ` U. x ) -> ( x e. y <-> x e. (TopOn ` U. x ) ) )
5		eleq1 2689	. . . . . . . 8 |- ( y = (TopOn ` U. x ) -> ( y e. ran TopOn <-> (TopOn ` U. x ) e. ran TopOn ) )
6	4, 5	anbi12d 747	. . . . . . 7 |- ( y = (TopOn ` U. x ) -> ( ( x e. y /\ y e. ran TopOn ) <-> ( x e. (TopOn ` U. x ) /\ (TopOn ` U. x ) e. ran TopOn ) ) )
7		simpl 473	. . . . . . . . 9 |- ( ( x e. (TopOn ` U. x ) /\ (TopOn ` U. x ) e. ran TopOn ) -> x e. (TopOn ` U. x ) )
8		fntopon 20728	. . . . . . . . . . . 12 |- TopOn Fn _V
9		vuniex 6954	. . . . . . . . . . . 12 |- U. x e. _V
10	8, 9	pm3.2i 471	. . . . . . . . . . 11 |- (TopOn Fn _V /\ U. x e. _V )
11		fnfvelrn 6356	. . . . . . . . . . 11 |- ( (TopOn Fn _V /\ U. x e. _V ) -> (TopOn ` U. x ) e. ran TopOn )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- (TopOn ` U. x ) e. ran TopOn
13	12	jctr 565	. . . . . . . . 9 |- ( x e. (TopOn ` U. x ) -> ( x e. (TopOn ` U. x ) /\ (TopOn ` U. x ) e. ran TopOn ) )
14	7, 13	impbii 199	. . . . . . . 8 |- ( ( x e. (TopOn ` U. x ) /\ (TopOn ` U. x ) e. ran TopOn ) <-> x e. (TopOn ` U. x ) )
15	14	a1i 11	. . . . . . 7 |- ( y = (TopOn ` U. x ) -> ( ( x e. (TopOn ` U. x ) /\ (TopOn ` U. x ) e. ran TopOn ) <-> x e. (TopOn ` U. x ) ) )
16	6, 15	bitrd 268	. . . . . 6 |- ( y = (TopOn ` U. x ) -> ( ( x e. y /\ y e. ran TopOn ) <-> x e. (TopOn ` U. x ) ) )
17	3, 16	spcev 3300	. . . . 5 |- ( x e. (TopOn ` U. x ) -> E. y ( x e. y /\ y e. ran TopOn ) )
18	2, 17	syl 17	. . . 4 |- ( x e. Top -> E. y ( x e. y /\ y e. ran TopOn ) )
19		funtopon 20725	. . . . . . . . . 10 |- Fun TopOn
20		elrnrexdm 6363	. . . . . . . . . 10 |- ( Fun TopOn -> ( y e. ran TopOn -> E. z e. dom TopOn y = (TopOn ` z ) ) )
21	19, 20	ax-mp 5	. . . . . . . . 9 |- ( y e. ran TopOn -> E. z e. dom TopOn y = (TopOn ` z ) )
22		rexex 3002	. . . . . . . . 9 |- ( E. z e. dom TopOn y = (TopOn ` z ) -> E. z y = (TopOn ` z ) )
23	21, 22	syl 17	. . . . . . . 8 |- ( y e. ran TopOn -> E. z y = (TopOn ` z ) )
24	23	anim2i 593	. . . . . . 7 |- ( ( x e. y /\ y e. ran TopOn ) -> ( x e. y /\ E. z y = (TopOn ` z ) ) )
25		19.42v 1918	. . . . . . . . 9 |- ( E. z ( x e. y /\ y = (TopOn ` z ) ) <-> ( x e. y /\ E. z y = (TopOn ` z ) ) )
26	25	biimpri 218	. . . . . . . 8 |- ( ( x e. y /\ E. z y = (TopOn ` z ) ) -> E. z ( x e. y /\ y = (TopOn ` z ) ) )
27		eqimss 3657	. . . . . . . . . . . 12 |- ( y = (TopOn ` z ) -> y C_ (TopOn ` z ) )
28	27	sseld 3602	. . . . . . . . . . 11 |- ( y = (TopOn ` z ) -> ( x e. y -> x e. (TopOn ` z ) ) )
29	28	com12 32	. . . . . . . . . 10 |- ( x e. y -> ( y = (TopOn ` z ) -> x e. (TopOn ` z ) ) )
30	29	imp 445	. . . . . . . . 9 |- ( ( x e. y /\ y = (TopOn ` z ) ) -> x e. (TopOn ` z ) )
31	30	eximi 1762	. . . . . . . 8 |- ( E. z ( x e. y /\ y = (TopOn ` z ) ) -> E. z x e. (TopOn ` z ) )
32	26, 31	syl 17	. . . . . . 7 |- ( ( x e. y /\ E. z y = (TopOn ` z ) ) -> E. z x e. (TopOn ` z ) )
33	24, 32	syl 17	. . . . . 6 |- ( ( x e. y /\ y e. ran TopOn ) -> E. z x e. (TopOn ` z ) )
34		topontop 20718	. . . . . . . 8 |- ( x e. (TopOn ` z ) -> x e. Top )
35	34	eximi 1762	. . . . . . 7 |- ( E. z x e. (TopOn ` z ) -> E. z x e. Top )
36		ax5e 1841	. . . . . . 7 |- ( E. z x e. Top -> x e. Top )
37	35, 36	syl 17	. . . . . 6 |- ( E. z x e. (TopOn ` z ) -> x e. Top )
38	33, 37	syl 17	. . . . 5 |- ( ( x e. y /\ y e. ran TopOn ) -> x e. Top )
39	38	exlimiv 1858	. . . 4 |- ( E. y ( x e. y /\ y e. ran TopOn ) -> x e. Top )
40	18, 39	impbii 199	. . 3 |- ( x e. Top <-> E. y ( x e. y /\ y e. ran TopOn ) )
41		eluni 4439	. . . 4 |- ( x e. U. ran TopOn <-> E. y ( x e. y /\ y e. ran TopOn ) )
42	41	bicomi 214	. . 3 |- ( E. y ( x e. y /\ y e. ran TopOn ) <-> x e. U. ran TopOn )
43	40, 42	bitri 264	. 2 |- ( x e. Top <-> x e. U. ran TopOn )
44	43	eqriv 2619	1 |- Top = U. ran TopOn

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   U.cuni 4436   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-topon 20716

This theorem is referenced by: (None)