Theorem tpsex 7547

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 4571) 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 2610	. . 3 |- (ATopSpJ <-> <.A, J>. e. TopSp)
2		relopab 3256	. . . . 5 |- Rel {<.x, y>. | (y e. Top /\ x = U.y)}
3		df-topsp 7535	. . . . . 6 |- TopSp = {<.x, y>. | (y e. Top /\ x = U.y)}
4	3	releqi 3234	. . . . 5 |- (Rel TopSp <-> Rel {<.x, y>. | (y e. Top /\ x = U.y)})
5	2, 4	mpbir 190	. . . 4 |- Rel TopSp
6	5	brrelexi 3198	. . 3 |- (ATopSpJ -> A e. V)
7	1, 6	sylbir 201	. 2 |- (<.A, J>. e. TopSp -> A e. V)
8		elirr 4571	. . . . . 6 |- -. A e. A
9		pm3.27 323	. . . . . . 7 |- ((A e. Top /\ A = U.A) -> A = U.A)
10		eqid 1468	. . . . . . . . 9 |- U.A = U.A
11	10	topopn 7544	. . . . . . . 8 |- (A e. Top -> U.A e. A)
12	11	adantr 389	. . . . . . 7 |- ((A e. Top /\ A = U.A) -> U.A e. A)
13	9, 12	eqeltrd 1540	. . . . . 6 |- ((A e. Top /\ A = U.A) -> A e. A)
14	8, 13	mto 106	. . . . 5 |- -. (A e. Top /\ A = U.A)
15		df-br 2610	. . . . . . . 8 |- (ATopSpA <-> <.A, A>. e. TopSp)
16	5	brrelexi 3198	. . . . . . . 8 |- (ATopSpA -> A e. V)
17	15, 16	sylbir 201	. . . . . . 7 |- (<.A, A>. e. TopSp -> A e. V)
18	17, 17	jca 288	. . . . . 6 |- (<.A, A>. e. TopSp -> (A e. V /\ A e. V))
19		elisset 1808	. . . . . . . 8 |- (A e. Top -> A e. V)
20	19, 19	jca 288	. . . . . . 7 |- (A e. Top -> (A e. V /\ A e. V))
21	20	adantr 389	. . . . . 6 |- ((A e. Top /\ A = U.A) -> (A e. V /\ A e. V))
22		eqeq1 1473	. . . . . . . . 9 |- (x = A -> (x = U.y <-> A = U.y))
23	22	anbi2d 614	. . . . . . . 8 |- (x = A -> ((y e. Top /\ x = U.y) <-> (y e. Top /\ A = U.y)))
24		eleq1 1526	. . . . . . . . 9 |- (y = A -> (y e. Top <-> A e. Top))
25		unieq 2500	. . . . . . . . . 10 |- (y = A -> U.y = U.A)
26	25	eqeq2d 1478	. . . . . . . . 9 |- (y = A -> (A = U.y <-> A = U.A))
27	24, 26	anbi12d 626	. . . . . . . 8 |- (y = A -> ((y e. Top /\ A = U.y) <-> (A e. Top /\ A = U.A)))
28	23, 27	opelopabg 2806	. . . . . . 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 1530	. . . . . . 7 |- (<.A, A>. e. TopSp <-> <.A, A>. e. {<.x, y>. | (y e. Top /\ x = U.y)})
30	28, 29	syl5bb 530	. . . . . 6 |- ((A e. V /\ A e. V) -> (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A)))
31	18, 21, 30	pm5.21nii 677	. . . . 5 |- (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A))
32	14, 31	mtbir 192	. . . 4 |- -. <.A, A>. e. TopSp
33		opprc2 2490	. . . . 5 |- (-. J e. V -> <.A, J>. = <.A, A>.)
34	33	eleq1d 1532	. . . 4 |- (-. J e. V -> (<.A, J>. e. TopSp <-> <.A, A>. e. TopSp))
35	32, 34	mtbiri 715	. . 3 |- (-. J e. V -> -. <.A, J>. e. TopSp)
36	35	a3i 74	. 2 |- (<.A, J>. e. TopSp -> J e. V)
37	7, 36	jca 288	1 |- (<.A, J>. e. TopSp -> (A e. V /\ J e. V))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  <.cop 2401  U.cuni 2493   class class class wbr 2609  {copab 2656  Rel wrel 3165  Topctop 7530  TopSpctps 7531

This theorem is referenced by:  istps 7548

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-reg 4565

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-rel 3175  df-top 7534  df-topsp 7535

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