Theorem homeofval 10516

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 2721	. . 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 3689	. . . . . . 7 |- (f:X-1-1-onto->Y -> f:X-->Y)
3	2	3ad2ant1 800	. . . . . 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 2116	. . . 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 4328	. . . 4 |- ((X e. V /\ Y e. V) -> {f | f:X-->Y} e. V)
9		uniexg 2871	. . . . 5 |- (J e. Top -> U.J e. V)
10		homeofval.1	. . . . 5 |- X = U.J
11	9, 10	syl5eqel 1552	. . . 4 |- (J e. Top -> X e. V)
12		uniexg 2871	. . . . 5 |- (K e. Top -> U.K e. V)
13		homeofval.2	. . . . 5 |- Y = U.K
14	12, 13	syl5eqel 1552	. . . 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 2510	. . . . . . . 8 |- (j = J -> U.j = U.J)
18	17, 10	syl6eqr 1525	. . . . . . 7 |- (j = J -> U.j = X)
19		f1oeq2 3685	. . . . . . 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 1786	. . . . . 6 |- (j = J -> (A.x e. j (f"x) e. k <-> A.x e. J (f"x) e. k))
22		eleq2 1535	. . . . . . 7 |- (j = J -> ((`'f"x) e. j <-> (`'f"x) e. J))
23	22	ralbidv 1663	. . . . . 6 |- (j = J -> (A.x e. k (`'f"x) e. j <-> A.x e. k (`'f"x) e. J))
24	20, 21, 23	3anbi123d 893	. . . . 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 1577	. . . 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 2510	. . . . . . . 8 |- (k = K -> U.k = U.K)
27	26, 13	syl6eqr 1525	. . . . . . 7 |- (k = K -> U.k = Y)
28		f1oeq3 3686	. . . . . . 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 1535	. . . . . . 7 |- (k = K -> ((f"x) e. k <-> (f"x) e. K))
31	30	ralbidv 1663	. . . . . 6 |- (k = K -> (A.x e. J (f"x) e. k <-> A.x e. J (f"x) e. K))
32		raleq1 1786	. . . . . 6 |- (k = K -> (A.x e. k (`'f"x) e. J <-> A.x e. K (`'f"x) e. J))
33	29, 31, 32	3anbi123d 893	. . . . 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 1577	. . . 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 10515	. . . 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 4027	. . 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 833	. 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 704	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 775  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811   (_ wss 2047  U.cuni 2503  `'ccnv 3169  "cima 3173  -->wf 3178  -1-1-onto->wf1o 3181  (class class class)co 3963  Topctop 7588   Homeo chomeosm 10513

This theorem is referenced by:  ishomeo 10517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-homeo 10515

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