Theorem homeofval 10234

Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 20-Feb-2007.)

Hypotheses

Ref	Expression
homeofval.1	|- X = U.J
homeofval.2	|- Y = U.K

Assertion

Ref	Expression
homeofval	|- ((J e. Top /\ K e. Top) -> (J Homeo K) = {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)})

Distinct variable groups:   f,J,x   f,K,x   f,X   f,Y

Proof of Theorem homeofval

Step	Hyp	Ref	Expression
1		f1of 4635	. . . . . 6 |- (f:X-1-1-onto->Y -> f:X-->Y)
2	1	3ad2ant1 897	. . . . 5 |- ((f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J) -> f:X-->Y)
3	2	a1i 8	. . . 4 |- ((J e. Top /\ K e. Top) -> ((f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J) -> f:X-->Y))
4	3	ss2abdv 2680	. . 3 |- ((J e. Top /\ K e. Top) -> {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} C_ {f | f:X-->Y})
5		mapex 5387	. . . 4 |- ((X e. _V /\ Y e. _V) -> {f | f:X-->Y} e. _V)
6		uniexg 3795	. . . . 5 |- (J e. Top -> U.J e. _V)
7		homeofval.1	. . . . 5 |- X = U.J
8	6, 7	syl5eqel 1975	. . . 4 |- (J e. Top -> X e. _V)
9		uniexg 3795	. . . . 5 |- (K e. Top -> U.K e. _V)
10		homeofval.2	. . . . 5 |- Y = U.K
11	9, 10	syl5eqel 1975	. . . 4 |- (K e. Top -> Y e. _V)
12	5, 8, 11	syl2an 503	. . 3 |- ((J e. Top /\ K e. Top) -> {f | f:X-->Y} e. _V)
13		ssexg 3457	. . 3 |- (({f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} C_ {f | f:X-->Y} /\ {f | f:X-->Y} e. _V) -> {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} e. _V)
14	4, 12, 13	syl11anc 524	. 2 |- ((J e. Top /\ K e. Top) -> {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} e. _V)
15		unieq 3185	. . . . . . 7 |- (j = J -> U.j = U.J)
16	15, 7	syl6eqr 1946	. . . . . 6 |- (j = J -> U.j = X)
17		f1oeq2 4631	. . . . . 6 |- (U.j = X -> (f:U.j-1-1-onto->U.k <-> f:X-1-1-onto->U.k))
18	16, 17	syl 12	. . . . 5 |- (j = J -> (f:U.j-1-1-onto->U.k <-> f:X-1-1-onto->U.k))
19		raleq 2266	. . . . 5 |- (j = J -> (A.x e. j (f"x) e. k <-> A.x e. J (f"x) e. k))
20		eleq2 1958	. . . . . 6 |- (j = J -> ((`'f"x) e. j <-> (`'f"x) e. J))
21	20	ralbidv 2123	. . . . 5 |- (j = J -> (A.x e. k (`'f"x) e. j <-> A.x e. k (`'f"x) e. J))
22	18, 19, 21	3anbi123d 1168	. . . 4 |- (j = J -> ((f:U.j-1-1-onto->U.k /\ A.x e. j (f"x) e. k /\ A.x e. k (`'f"x) e. j) <-> (f:X-1-1-onto->U.k /\ A.x e. J (f"x) e. k /\ A.x e. k (`'f"x) e. J)))
23	22	abbidv 2008	. . 3 |- (j = J -> {f | (f:U.j-1-1-onto->U.k /\ A.x e. j (f"x) e. k /\ A.x e. k (`'f"x) e. j)} = {f | (f:X-1-1-onto->U.k /\ A.x e. J (f"x) e. k /\ A.x e. k (`'f"x) e. J)})
24		unieq 3185	. . . . . . 7 |- (k = K -> U.k = U.K)
25	24, 10	syl6eqr 1946	. . . . . 6 |- (k = K -> U.k = Y)
26		f1oeq3 4632	. . . . . 6 |- (U.k = Y -> (f:X-1-1-onto->U.k <-> f:X-1-1-onto->Y))
27	25, 26	syl 12	. . . . 5 |- (k = K -> (f:X-1-1-onto->U.k <-> f:X-1-1-onto->Y))
28		eleq2 1958	. . . . . 6 |- (k = K -> ((f"x) e. k <-> (f"x) e. K))
29	28	ralbidv 2123	. . . . 5 |- (k = K -> (A.x e. J (f"x) e. k <-> A.x e. J (f"x) e. K))
30		raleq 2266	. . . . 5 |- (k = K -> (A.x e. k (`'f"x) e. J <-> A.x e. K (`'f"x) e. J))
31	27, 29, 30	3anbi123d 1168	. . . 4 |- (k = K -> ((f:X-1-1-onto->U.k /\ A.x e. J (f"x) e. k /\ A.x e. k (`'f"x) e. J) <-> (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)))
32	31	abbidv 2008	. . 3 |- (k = K -> {f | (f:X-1-1-onto->U.k /\ A.x e. J (f"x) e. k /\ A.x e. k (`'f"x) e. J)} = {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)})
33		df-homeo 10232	. . 3 |- Homeo = {<.<.j, k>., z>. | ((j e. Top /\ k e. Top) /\ z = {f | (f:U.j-1-1-onto->U.k /\ A.x e. j (f"x) e. k /\ A.x e. k (`'f"x) e. j)})}
34	23, 32, 33	oprabval2g 4956	. 2 |- ((J e. Top /\ K e. Top /\ {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} e. _V) -> (J Homeo K) = {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)})
35	14, 34	mpd3an3 1192	1 |- ((J e. Top /\ K e. Top) -> (J Homeo K) = {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593  U.cuni 3177  `'ccnv 3985  "cima 3989  -->wf 3994  -1-1-onto->wf1o 3997  (class class class)co 4884  Topctop 8857   Homeo chomeosm 10230

This theorem is referenced by:  ishomeo 10235

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-v 2294  df-sbc 2454  df-csb 2541  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-opr 4886  df-oprab 4887  df-homeo 10232

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