Theorem homeofval 10497

Description: The set of all the homeomorphisms between two topologies.

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		ssexg 2718	. . 3 |- (({f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} (_ {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)
2		f1of 3686	. . . . . . 7 |- (f:X-1-1-onto->Y -> f:X-->Y)
3	2	3ad2ant1 799	. . . . . 6 |- ((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	a1i 8	. . . . 5 |- ((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))
5	4	19.21aiv 1286	. . . 4 |- ((J e. Top /\ K e. Top) -> A.f((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))
6		ss2ab 2114	. . . 4 |- ({f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)} (_ {f | f:X-->Y} <-> A.f((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))
7	5, 6	sylibr 200	. . 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)} (_ {f | f:X-->Y})
8		mapex 4325	. . . 4 |- ((X e. V /\ Y e. V) -> {f | f:X-->Y} e. V)
9		uniexg 2868	. . . . 5 |- (J e. Top -> U.J e. V)
10		homeofval.1	. . . . 5 |- X = U.J
11	9, 10	syl5eqel 1551	. . . 4 |- (J e. Top -> X e. V)
12		uniexg 2868	. . . . 5 |- (K e. Top -> U.K e. V)
13		homeofval.2	. . . . 5 |- Y = U.K
14	12, 13	syl5eqel 1551	. . . 4 |- (K e. Top -> Y e. V)
15	8, 11, 14	syl2an 454	. . 3 |- ((J e. Top /\ K e. Top) -> {f | f:X-->Y} e. V)
16	1, 7, 15	sylanc 471	. 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)
17		unieq 2507	. . . . . . . 8 |- (j = J -> U.j = U.J)
18	17, 10	syl6eqr 1524	. . . . . . 7 |- (j = J -> U.j = X)
19		f1oeq2 3682	. . . . . . 7 |- (U.j = X -> (f:U.j-1-1-onto->U.k <-> f:X-1-1-onto->U.k))
20	18, 19	syl 10	. . . . . 6 |- (j = J -> (f:U.j-1-1-onto->U.k <-> f:X-1-1-onto->U.k))
21		raleq1 1785	. . . . . 6 |- (j = J -> (A.x e. j (f"x) e. k <-> A.x e. J (f"x) e. k))
22		eleq2 1534	. . . . . . 7 |- (j = J -> ((`'f"x) e. j <-> (`'f"x) e. J))
23	22	ralbidv 1662	. . . . . 6 |- (j = J -> (A.x e. k (`'f"x) e. j <-> A.x e. k (`'f"x) e. J))
24	20, 21, 23	3anbi123d 892	. . . . 5 |- (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)))
25	24	abbidv 1576	. . . 4 |- (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)})
26		unieq 2507	. . . . . . . 8 |- (k = K -> U.k = U.K)
27	26, 13	syl6eqr 1524	. . . . . . 7 |- (k = K -> U.k = Y)
28		f1oeq3 3683	. . . . . . 7 |- (U.k = Y -> (f:X-1-1-onto->U.k <-> f:X-1-1-onto->Y))
29	27, 28	syl 10	. . . . . 6 |- (k = K -> (f:X-1-1-onto->U.k <-> f:X-1-1-onto->Y))
30		eleq2 1534	. . . . . . 7 |- (k = K -> ((f"x) e. k <-> (f"x) e. K))
31	30	ralbidv 1662	. . . . . 6 |- (k = K -> (A.x e. J (f"x) e. k <-> A.x e. J (f"x) e. K))
32		raleq1 1785	. . . . . 6 |- (k = K -> (A.x e. k (`'f"x) e. J <-> A.x e. K (`'f"x) e. J))
33	29, 31, 32	3anbi123d 892	. . . . 5 |- (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)))
34	33	abbidv 1576	. . . 4 |- (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)})
35		df-homeo 10496	. . . 4 |- 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)})}
36	25, 34, 35	oprabval2g 4024	. . 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)} 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)})
37	36	3expa 832	. 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)})
38	16, 37	mpdan 703	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 146   /\ wa 223   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957  {cab 1463  A.wral 1644  Vcvv 1809   (_ wss 2045  U.cuni 2500  `'ccnv 3166  "cima 3170  -->wf 3175  -1-1-onto->wf1o 3178  (class class class)co 3960  Topctop 7567   Homeo chomeosm 10494

This theorem is referenced by:  ishomeo 10498

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-9 964  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 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-opr 3962  df-oprab 3963  df-homeo 10496

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