Theorem tpsex 8874

Description: Existence implied by membership in a topological space. This technical lemma, which can help eliminate unnecessary antecedents, uses the Axiom of Regularity (via elirr 5701) along with definitional tricks.

Assertion

Ref	Expression
tpsex	|- (<.A, J>. e. TopSp -> (A e. _V /\ J e. _V))

Proof of Theorem tpsex

Step	Hyp	Ref	Expression
1		df-br 3339	. . 3 |- (ATopSpJ <-> <.A, J>. e. TopSp)
2		relopab 4104	. . . . 5 |- Rel {<.x, y>. | (y e. Top /\ x = U.y)}
3		df-topsp 8862	. . . . . 6 |- TopSp = {<.x, y>. | (y e. Top /\ x = U.y)}
4	3	releqi 4072	. . . . 5 |- (Rel TopSp <-> Rel {<.x, y>. | (y e. Top /\ x = U.y)})
5	2, 4	mpbir 207	. . . 4 |- Rel TopSp
6	5	brrelexi 4029	. . 3 |- (ATopSpJ -> A e. _V)
7	1, 6	sylbir 218	. 2 |- (<.A, J>. e. TopSp -> A e. _V)
8		elirr 5701	. . . . . 6 |- -. A e. A
9		simpr 350	. . . . . . 7 |- ((A e. Top /\ A = U.A) -> A = U.A)
10		eqid 1884	. . . . . . . . 9 |- U.A = U.A
11	10	topopn 8871	. . . . . . . 8 |- (A e. Top -> U.A e. A)
12	11	adantr 425	. . . . . . 7 |- ((A e. Top /\ A = U.A) -> U.A e. A)
13	9, 12	eqeltrd 1971	. . . . . 6 |- ((A e. Top /\ A = U.A) -> A e. A)
14	8, 13	mto 121	. . . . 5 |- -. (A e. Top /\ A = U.A)
15		df-br 3339	. . . . . . . 8 |- (ATopSpA <-> <.A, A>. e. TopSp)
16	5	brrelexi 4029	. . . . . . . 8 |- (ATopSpA -> A e. _V)
17	15, 16	sylbir 218	. . . . . . 7 |- (<.A, A>. e. TopSp -> A e. _V)
18	17, 17	jca 310	. . . . . 6 |- (<.A, A>. e. TopSp -> (A e. _V /\ A e. _V))
19		elisset 2299	. . . . . . . 8 |- (A e. Top -> A e. _V)
20	19, 19	jca 310	. . . . . . 7 |- (A e. Top -> (A e. _V /\ A e. _V))
21	20	adantr 425	. . . . . 6 |- ((A e. Top /\ A = U.A) -> (A e. _V /\ A e. _V))
22		eqeq1 1890	. . . . . . . . 9 |- (x = A -> (x = U.y <-> A = U.y))
23	22	anbi2d 678	. . . . . . . 8 |- (x = A -> ((y e. Top /\ x = U.y) <-> (y e. Top /\ A = U.y)))
24		eleq1 1957	. . . . . . . . 9 |- (y = A -> (y e. Top <-> A e. Top))
25		unieq 3185	. . . . . . . . . 10 |- (y = A -> U.y = U.A)
26	25	eqeq2d 1895	. . . . . . . . 9 |- (y = A -> (A = U.y <-> A = U.A))
27	24, 26	anbi12d 690	. . . . . . . 8 |- (y = A -> ((y e. Top /\ A = U.y) <-> (A e. Top /\ A = U.A)))
28	23, 27	opelopabg 3567	. . . . . . 7 |- ((A e. _V /\ A e. _V) -> (<.A, A>. e. {<.x, y>. | (y e. Top /\ x = U.y)} <-> (A e. Top /\ A = U.A)))
29	3	eleq2i 1961	. . . . . . 7 |- (<.A, A>. e. TopSp <-> <.A, A>. e. {<.x, y>. | (y e. Top /\ x = U.y)})
30	28, 29	syl5bb 591	. . . . . 6 |- ((A e. _V /\ A e. _V) -> (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A)))
31	18, 21, 30	pm5.21nii 743	. . . . 5 |- (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A))
32	14, 31	mtbir 209	. . . 4 |- -. <.A, A>. e. TopSp
33		opprc2 3171	. . . . 5 |- (-. J e. _V -> <.A, J>. = <.A, A>.)
34	33	eleq1d 1963	. . . 4 |- (-. J e. _V -> (<.A, J>. e. TopSp <-> <.A, A>. e. TopSp))
35	32, 34	mtbiri 785	. . 3 |- (-. J e. _V -> -. <.A, J>. e. TopSp)
36	35	con4i 90	. 2 |- (<.A, J>. e. TopSp -> J e. _V)
37	7, 36	jca 310	1 |- (<.A, J>. e. TopSp -> (A e. _V /\ J e. _V))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395  Rel wrel 3991  Topctop 8857  TopSpctps 8858

This theorem is referenced by:  istps 8875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-top 8861  df-topsp 8862

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