Theorem tpsex 7565

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 4582) 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 2616	. . 3 |- (ATopSpJ <-> <.A, J>. e. TopSp)
2		relopab 3262	. . . . 5 |- Rel {<.x, y>. | (y e. Top /\ x = U.y)}
3		df-topsp 7553	. . . . . 6 |- TopSp = {<.x, y>. | (y e. Top /\ x = U.y)}
4	3	releqi 3240	. . . . 5 |- (Rel TopSp <-> Rel {<.x, y>. | (y e. Top /\ x = U.y)})
5	2, 4	mpbir 190	. . . 4 |- Rel TopSp
6	5	brrelexi 3204	. . 3 |- (ATopSpJ -> A e. V)
7	1, 6	sylbir 201	. 2 |- (<.A, J>. e. TopSp -> A e. V)
8		elirr 4582	. . . . . 6 |- -. A e. A
9		pm3.27 323	. . . . . . 7 |- ((A e. Top /\ A = U.A) -> A = U.A)
10		eqid 1474	. . . . . . . . 9 |- U.A = U.A
11	10	topopn 7562	. . . . . . . 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 1546	. . . . . 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 2616	. . . . . . . 8 |- (ATopSpA <-> <.A, A>. e. TopSp)
16	5	brrelexi 3204	. . . . . . . 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 1814	. . . . . . . 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 1479	. . . . . . . . 9 |- (x = A -> (x = U.y <-> A = U.y))
23	22	anbi2d 615	. . . . . . . 8 |- (x = A -> ((y e. Top /\ x = U.y) <-> (y e. Top /\ A = U.y)))
24		eleq1 1532	. . . . . . . . 9 |- (y = A -> (y e. Top <-> A e. Top))
25		unieq 2506	. . . . . . . . . 10 |- (y = A -> U.y = U.A)
26	25	eqeq2d 1484	. . . . . . . . 9 |- (y = A -> (A = U.y <-> A = U.A))
27	24, 26	anbi12d 627	. . . . . . . 8 |- (y = A -> ((y e. Top /\ A = U.y) <-> (A e. Top /\ A = U.A)))
28	23, 27	opelopabg 2813	. . . . . . 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 1536	. . . . . . 7 |- (<.A, A>. e. TopSp <-> <.A, A>. e. {<.x, y>. | (y e. Top /\ x = U.y)})
30	28, 29	syl5bb 531	. . . . . 6 |- ((A e. V /\ A e. V) -> (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A)))
31	18, 21, 30	pm5.21nii 678	. . . . 5 |- (<.A, A>. e. TopSp <-> (A e. Top /\ A = U.A))
32	14, 31	mtbir 192	. . . 4 |- -. <.A, A>. e. TopSp
33		opprc2 2496	. . . . 5 |- (-. J e. V -> <.A, J>. = <.A, A>.)
34	33	eleq1d 1538	. . . 4 |- (-. J e. V -> (<.A, J>. e. TopSp <-> <.A, A>. e. TopSp))
35	32, 34	mtbiri 716	. . 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 955   e. wcel 957  Vcvv 1808  <.cop 2408  U.cuni 2499   class class class wbr 2615  {copab 2662  Rel wrel 3171  Topctop 7548  TopSpctps 7549

This theorem is referenced by:  istps 7566

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-reg 4576

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-rel 3181  df-top 7552  df-topsp 7553

Copyright terms: Public domain