Theorem opncldf3 15404

Description: The values of the converse/inverse of the open-closed bijection.

Hypotheses

Ref	Expression
opncldf.1	|- X = U.J
opncldf.2	|- F = {<.u, v>. | (u e. J /\ v = (X \ u))}

Assertion

Ref	Expression
opncldf3	|- ((J e. Top /\ B e. (Clsd` J)) -> (`'F` B) = (X \ B))

Distinct variable groups:   v,u,B   u,J,v   u,X,v

Proof of Theorem opncldf3

Step	Hyp	Ref	Expression
1		opncldf.1	. . . 4 |- X = U.J
2	1	cldopn 8948	. . 3 |- ((J e. Top /\ B e. (Clsd` J)) -> (X \ B) e. J)
3		opncldf.2	. . . . . 6 |- F = {<.u, v>. | (u e. J /\ v = (X \ u))}
4	1, 3	opncldf1 15402	. . . . 5 |- (J e. Top -> F:J-1-1-onto->(Clsd` J))
5		f1ocnv 4651	. . . . 5 |- (F:J-1-1-onto->(Clsd` J) -> `'F:(Clsd` J)-1-1-onto->J)
6		f1ofun 4637	. . . . 5 |- (`'F:(Clsd` J)-1-1-onto->J -> Fun `'F)
7	4, 5, 6	3syl 24	. . . 4 |- (J e. Top -> Fun `'F)
8	7	adantr 425	. . 3 |- ((J e. Top /\ B e. (Clsd` J)) -> Fun `'F)
9	2, 8	jca 310	. 2 |- ((J e. Top /\ B e. (Clsd` J)) -> ((X \ B) e. J /\ Fun `'F))
10		eleq1 1957	. . . . . . . 8 |- (u = (X \ B) -> (u e. J <-> (X \ B) e. J))
11		difeq2 2719	. . . . . . . . 9 |- (u = (X \ B) -> (X \ u) = (X \ (X \ B)))
12	11	eqeq2d 1895	. . . . . . . 8 |- (u = (X \ B) -> (v = (X \ u) <-> v = (X \ (X \ B))))
13	10, 12	anbi12d 690	. . . . . . 7 |- (u = (X \ B) -> ((u e. J /\ v = (X \ u)) <-> ((X \ B) e. J /\ v = (X \ (X \ B)))))
14		eqeq1 1890	. . . . . . . 8 |- (v = B -> (v = (X \ (X \ B)) <-> B = (X \ (X \ B))))
15	14	anbi2d 678	. . . . . . 7 |- (v = B -> (((X \ B) e. J /\ v = (X \ (X \ B))) <-> ((X \ B) e. J /\ B = (X \ (X \ B)))))
16	13, 15	opelopabg 3567	. . . . . 6 |- (((X \ B) e. J /\ B e. (Clsd` J)) -> (<.(X \ B), B>. e. {<.u, v>. | (u e. J /\ v = (X \ u))} <-> ((X \ B) e. J /\ B = (X \ (X \ B)))))
17	2, 16	sylancom 531	. . . . 5 |- ((J e. Top /\ B e. (Clsd` J)) -> (<.(X \ B), B>. e. {<.u, v>. | (u e. J /\ v = (X \ u))} <-> ((X \ B) e. J /\ B = (X \ (X \ B)))))
18	3	eleq2i 1961	. . . . 5 |- (<.(X \ B), B>. e. F <-> <.(X \ B), B>. e. {<.u, v>. | (u e. J /\ v = (X \ u))})
19	17, 18	syl5bb 591	. . . 4 |- ((J e. Top /\ B e. (Clsd` J)) -> (<.(X \ B), B>. e. F <-> ((X \ B) e. J /\ B = (X \ (X \ B)))))
20	1	cldss 8947	. . . . 5 |- ((J e. Top /\ B e. (Clsd` J)) -> B C_ X)
21		dfss4 2827	. . . . . 6 |- (B C_ X <-> (X \ (X \ B)) = B)
22		eqcom 1886	. . . . . 6 |- ((X \ (X \ B)) = B <-> B = (X \ (X \ B)))
23	21, 22	bitri 190	. . . . 5 |- (B C_ X <-> B = (X \ (X \ B)))
24	20, 23	sylib 215	. . . 4 |- ((J e. Top /\ B e. (Clsd` J)) -> B = (X \ (X \ B)))
25	19, 2, 24	mpbir2and 802	. . 3 |- ((J e. Top /\ B e. (Clsd` J)) -> <.(X \ B), B>. e. F)
26		simpr 350	. . . 4 |- ((J e. Top /\ B e. (Clsd` J)) -> B e. (Clsd` J))
27		opelcnvg 4140	. . . 4 |- ((B e. (Clsd` J) /\ (X \ B) e. J) -> (<.B, (X \ B)>. e. `'F <-> <.(X \ B), B>. e. F))
28	26, 2, 27	syl11anc 524	. . 3 |- ((J e. Top /\ B e. (Clsd` J)) -> (<.B, (X \ B)>. e. `'F <-> <.(X \ B), B>. e. F))
29	25, 28	mpbird 213	. 2 |- ((J e. Top /\ B e. (Clsd` J)) -> <.B, (X \ B)>. e. `'F)
30		funopfvg 4711	. 2 |- (((X \ B) e. J /\ Fun `'F) -> (<.B, (X \ B)>. e. `'F -> (`'F` B) = (X \ B)))
31	9, 29, 30	sylc 83	1 |- ((J e. Top /\ B e. (Clsd` J)) -> (`'F` B) = (X \ B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   C_ wss 2593  <.cop 3046  U.cuni 3177  {copab 3395  `'ccnv 3985  Fun wfun 3992  -1-1-onto->wf1o 3997  ` cfv 3998  Topctop 8857  Clsdccld 8936

This theorem is referenced by:  compfipin0lem 15435  compfipin0 15436

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-13 1311  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-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-rab 2112  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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-cld 8939

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