Theorem ishomeo 10498

Description: The predicate F is a homeomorphism between topology J and topology K. Based on Bourbaki TG I.2.

Hypotheses

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

Assertion

Ref	Expression
ishomeo	|- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> (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:   x,F   x,J   x,K

Proof of Theorem ishomeo

Step	Hyp	Ref	Expression
1		ishomeo.1	. . . . 5 |- X = U.J
2		ishomeo.2	. . . . 5 |- Y = U.K
3	1, 2	homeofval 10497	. . . 4 |- ((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)})
4	3	3adant3 798	. . 3 |- ((J e. Top /\ K e. Top /\ F e. A) -> (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)})
5	4	eleq2d 1540	. 2 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> F e. {f | (f:X-1-1-onto->Y /\ A.x e. J (f"x) e. K /\ A.x e. K (`'f"x) e. J)}))
6		f1oeq1 3681	. . . . 5 |- (f = F -> (f:X-1-1-onto->Y <-> F:X-1-1-onto->Y))
7		imaeq1 3398	. . . . . . 7 |- (f = F -> (f"x) = (F"x))
8	7	eleq1d 1539	. . . . . 6 |- (f = F -> ((f"x) e. K <-> (F"x) e. K))
9	8	ralbidv 1662	. . . . 5 |- (f = F -> (A.x e. J (f"x) e. K <-> A.x e. J (F"x) e. K))
10		cnveq 3289	. . . . . . . 8 |- (f = F -> `'f = `'F)
11	10	imaeq1d 3400	. . . . . . 7 |- (f = F -> (`'f"x) = (`'F"x))
12	11	eleq1d 1539	. . . . . 6 |- (f = F -> ((`'f"x) e. J <-> (`'F"x) e. J))
13	12	ralbidv 1662	. . . . 5 |- (f = F -> (A.x e. K (`'f"x) e. J <-> A.x e. K (`'F"x) e. J))
14	6, 9, 13	3anbi123d 892	. . . 4 |- (f = 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-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)))
15	14	elabg 1897	. . 3 |- (F e. A -> (F e. {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-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)))
16	15	3ad2ant3 801	. 2 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. {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-1-1-onto->Y /\ A.x e. J (F"x) e. K /\ A.x e. K (`'F"x) e. J)))
17	5, 16	bitrd 527	1 |- ((J e. Top /\ K e. Top /\ F e. A) -> (F e. (J Homeo K) <-> (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   /\ w3a 774   = wceq 955   e. wcel 957  {cab 1463  A.wral 1644  U.cuni 2500  `'ccnv 3166  "cima 3170  -1-1-onto->wf1o 3178  (class class class)co 3960  Topctop 7567   Homeo chomeosm 10494

This theorem is referenced by:  hmeomap 10499  hmeocna 10500  hmeocnb 10501  cmphmp 10502  idhme 10503  cnvhmpha 10506  cnvhmphb 10507  cnvhmph 10508  hmphsyma 10509  hmphre 10511  homcard 10520

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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