Theorem ishomeo 10503

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 10502	. . . 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 801	. . 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 1544	. 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 3690	. . . . 5 |- (f = F -> (f:X-1-1-onto->Y <-> F:X-1-1-onto->Y))
7		imaeq1 3407	. . . . . . 7 |- (f = F -> (f"x) = (F"x))
8	7	eleq1d 1543	. . . . . 6 |- (f = F -> ((f"x) e. K <-> (F"x) e. K))
9	8	ralbidv 1666	. . . . 5 |- (f = F -> (A.x e. J (f"x) e. K <-> A.x e. J (F"x) e. K))
10		cnveq 3298	. . . . . . . 8 |- (f = F -> `'f = `'F)
11	10	imaeq1d 3409	. . . . . . 7 |- (f = F -> (`'f"x) = (`'F"x))
12	11	eleq1d 1543	. . . . . 6 |- (f = F -> ((`'f"x) e. J <-> (`'F"x) e. J))
13	12	ralbidv 1666	. . . . 5 |- (f = F -> (A.x e. K (`'f"x) e. J <-> A.x e. K (`'F"x) e. J))
14	6, 9, 13	3anbi123d 895	. . . 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 1902	. . 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 804	. 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 530	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 777   = wceq 958   e. wcel 960  {cab 1466  A.wral 1648  U.cuni 2507  `'ccnv 3175  "cima 3179  -1-1-onto->wf1o 3187  (class class class)co 3969  Topctop 7590   Homeo chomeosm 10499

This theorem is referenced by:  hmeomap 10504  hmeocna 10505  hmeocnb 10506  cmphmp 10507  idhme 10508  cnvhmpha 10511  cnvhmphb 10512  cnvhmph 10513  hmphsyma 10514  hmphre 10516  homcard 10525  eqindhome 10527  hmeobc 10528

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-opr 3971  df-oprab 3972  df-homeo 10501

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